(* This file is a part of IsarMathLib - a library of formalized mathematics written for Isabelle/Isar. Copyright (C) 2009-2020 Slawomir Kolodynski This program is free software; Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE, DATA, OR PROFITS OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Topological groups - introduction› theory TopologicalGroup_ZF imports Topology_ZF_3 Group_ZF_1 Semigroup_ZF begin text‹This theory is about the first subject of algebraic topology: topological groups.› subsection‹Topological group: definition and notation› text‹Topological group is a group that is a topological space at the same time. This means that a topological group is a triple of sets, say $(G,f,T)$ such that $T$ is a topology on $G$, $f$ is a group operation on $G$ and both $f$ and the operation of taking inverse in $G$ are continuous. Since IsarMathLib defines topology without using the carrier, (see ‹Topology_ZF›), in our setup we just use $\bigcup T$ instead of $G$ and say that the pair of sets $(\bigcup T , f)$ is a group. This way our definition of being a topological group is a statement about two sets: the topology $T$ and the group operation $f$ on $G=\bigcup T$. Since the domain of the group operation is $G\times G$, the pair of topologies in which $f$ is supposed to be continuous is $T$ and the product topology on $G\times G$ (which we will call $\tau$ below).› text‹This way we arrive at the following definition of a predicate that states that pair of sets is a topological group.› definition "IsAtopologicalGroup(T,f) ≡ (T {is a topology}) ∧ IsAgroup(⋃T,f) ∧ IsContinuous(ProductTopology(T,T),T,f) ∧ IsContinuous(T,T,GroupInv(⋃T,f))" text‹We will inherit notation from the ‹topology0› locale. That locale assumes that $T$ is a topology. For convenience we will denote $G=\bigcup T$ and $\tau$ to be the product topology on $G\times G$. To that we add some notation specific to groups. We will use additive notation for the group operation, even though we don't assume that the group is abelian. The notation $g+A$ will mean the left translation of the set $A$ by element $g$, i.e. $g+A=\{g+a|a\in A\}$. The group operation $G$ induces a natural operation on the subsets of $G$ defined as $\langle A,B\rangle \mapsto \{x+y | x\in A, y\in B \}$. Such operation has been considered in ‹func_ZF› and called $f$ ''lifted to subsets of'' $G$. We will denote the value of such operation on sets $A,B$ as $A+B$. The set of neigboorhoods of zero (denoted ‹𝒩⇩_{0}›) is the collection of (not necessarily open) sets whose interior contains the neutral element of the group.› locale topgroup = topology0 + fixes G defines G_def [simp]: "G ≡ ⋃T" fixes prodtop ("τ") defines prodtop_def [simp]: "τ ≡ ProductTopology(T,T)" fixes f assumes Ggroup: "IsAgroup(G,f)" assumes fcon: "IsContinuous(τ,T,f)" assumes inv_cont: "IsContinuous(T,T,GroupInv(G,f))" fixes grop (infixl "\<ra>" 90) defines grop_def [simp]: "x\<ra>y ≡ f`⟨x,y⟩" fixes grinv ("\<rm> _" 89) defines grinv_def [simp]: "(\<rm>x) ≡ GroupInv(G,f)`(x)" fixes grsub (infixl "\<rs>" 90) defines grsub_def [simp]: "x\<rs>y ≡ x\<ra>(\<rm>y)" fixes setinv ("\<sm> _" 72) defines setninv_def [simp]: "\<sm>A ≡ GroupInv(G,f)``(A)" fixes ltrans (infix "\<ltr>" 73) defines ltrans_def [simp]: "x \<ltr> A ≡ LeftTranslation(G,f,x)``(A)" fixes rtrans (infix "\<rtr>" 73) defines rtrans_def [simp]: "A \<rtr> x ≡ RightTranslation(G,f,x)``(A)" fixes setadd (infixl "\<sad>" 71) defines setadd_def [simp]: "A\<sad>B ≡ (f {lifted to subsets of} G)`⟨A,B⟩" fixes gzero ("𝟬") defines gzero_def [simp]: "𝟬 ≡ TheNeutralElement(G,f)" fixes zerohoods ("𝒩⇩_{0}") defines zerohoods_def [simp]: "𝒩⇩_{0}≡ {A ∈ Pow(G). 𝟬 ∈ int(A)}" fixes listsum ("∑ _" 70) defines listsum_def[simp]: "∑k ≡ Fold1(f,k)" text‹The first lemma states that we indeeed talk about topological group in the context of ‹topgroup› locale.› lemma (in topgroup) topGroup: shows "IsAtopologicalGroup(T,f)" using topSpaceAssum Ggroup fcon inv_cont IsAtopologicalGroup_def by simp text‹If a pair of sets $(T,f)$ forms a topological group, then all theorems proven in the ‹topgroup› context are valid as applied to $(T,f)$.› lemma topGroupLocale: assumes "IsAtopologicalGroup(T,f)" shows "topgroup(T,f)" using assms IsAtopologicalGroup_def topgroup_def topgroup_axioms.intro topology0_def by simp text‹We can use the ‹group0› locale in the context of ‹topgroup›.› lemma (in topgroup) group0_valid_in_tgroup: shows "group0(G,f)" using Ggroup group0_def by simp text‹We can use the ‹group0› locale in the context of ‹topgroup›.› sublocale topgroup < group0 G f gzero grop grinv unfolding group0_def gzero_def grop_def grinv_def using Ggroup by auto text‹We can use ‹semigr0› locale in the context of ‹topgroup›.› lemma (in topgroup) semigr0_valid_in_tgroup: shows "semigr0(G,f)" using Ggroup IsAgroup_def IsAmonoid_def semigr0_def by simp text‹We can use the ‹prod_top_spaces0› locale in the context of ‹topgroup›.› lemma (in topgroup) prod_top_spaces0_valid: shows "prod_top_spaces0(T,T,T)" using topSpaceAssum prod_top_spaces0_def by simp text‹Negative of a group element is in group.› lemma (in topgroup) neg_in_tgroup: assumes "g∈G" shows "(\<rm>g) ∈ G" using assms inverse_in_group by simp text‹Sum of two group elements is in the group.› lemma (in topgroup) group_op_closed_add: assumes "x⇩_{1}∈ G" "x⇩_{2}∈ G" shows "x⇩_{1}\<ra>x⇩_{2}∈ G" using assms group_op_closed by simp text‹Zero is in the group.› lemma (in topgroup) zero_in_tgroup: shows "𝟬∈G" using group0_2_L2 by simp text‹ Another lemma about canceling with two group elements written in additive notation › lemma (in topgroup) inv_cancel_two_add: assumes "x⇩_{1}∈ G" "x⇩_{2}∈ G" shows "x⇩_{1}\<ra>(\<rm>x⇩_{2})\<ra>x⇩_{2}= x⇩_{1}" "x⇩_{1}\<ra>x⇩_{2}\<ra>(\<rm>x⇩_{2}) = x⇩_{1}" "(\<rm>x⇩_{1})\<ra>(x⇩_{1}\<ra>x⇩_{2}) = x⇩_{2}" "x⇩_{1}\<ra>((\<rm>x⇩_{1})\<ra>x⇩_{2}) = x⇩_{2}" using assms inv_cancel_two by auto text‹Useful identities proven in the ‹Group_ZF› theory, rewritten here in additive notation. Note since the group operation notation is left associative we don't really need the first set of parentheses in some cases.› lemma (in topgroup) cancel_middle_add: assumes "x⇩_{1}∈ G" "x⇩_{2}∈ G" "x⇩_{3}∈ G" shows "(x⇩_{1}\<ra>(\<rm>x⇩_{2}))\<ra>(x⇩_{2}\<ra>(\<rm>x⇩_{3})) = x⇩_{1}\<ra> (\<rm>x⇩_{3})" "((\<rm>x⇩_{1})\<ra>x⇩_{2})\<ra>((\<rm>x⇩_{2})\<ra>x⇩_{3}) = (\<rm>x⇩_{1})\<ra> x⇩_{3}" "(\<rm> (x⇩_{1}\<ra>x⇩_{2})) \<ra> (x⇩_{1}\<ra>x⇩_{3}) = (\<rm>x⇩_{2})\<ra>x⇩_{3}" "(x⇩_{1}\<ra>x⇩_{2}) \<ra> (\<rm>(x⇩_{3}\<ra>x⇩_{2})) =x⇩_{1}\<ra> (\<rm>x⇩_{3})" "(\<rm>x⇩_{1}) \<ra> (x⇩_{1}\<ra>x⇩_{2}\<ra>x⇩_{3}) \<ra> (\<rm>x⇩_{3}) = x⇩_{2}" proof - from assms have "x⇩_{1}\<ra> (\<rm>x⇩_{3}) = (x⇩_{1}\<ra>(\<rm>x⇩_{2}))\<ra>(x⇩_{2}\<ra>(\<rm>x⇩_{3}))" using group0_2_L14A(1) by blast thus "(x⇩_{1}\<ra>(\<rm>x⇩_{2}))\<ra>(x⇩_{2}\<ra>(\<rm>x⇩_{3})) = x⇩_{1}\<ra> (\<rm>x⇩_{3})" by simp from assms have "(\<rm>x⇩_{1})\<ra> x⇩_{3}= ((\<rm>x⇩_{1})\<ra>x⇩_{2})\<ra>((\<rm>x⇩_{2})\<ra>x⇩_{3})" using group0_2_L14A(2) by blast thus "((\<rm>x⇩_{1})\<ra>x⇩_{2})\<ra>((\<rm>x⇩_{2})\<ra>x⇩_{3}) = (\<rm>x⇩_{1})\<ra> x⇩_{3}" by simp from assms show "(\<rm> (x⇩_{1}\<ra>x⇩_{2})) \<ra> (x⇩_{1}\<ra>x⇩_{3}) = (\<rm>x⇩_{2})\<ra>x⇩_{3}" using cancel_middle(1) by simp from assms show "(x⇩_{1}\<ra>x⇩_{2}) \<ra> (\<rm>(x⇩_{3}\<ra>x⇩_{2})) =x⇩_{1}\<ra> (\<rm>x⇩_{3})" using cancel_middle(2) by simp from assms show "(\<rm>x⇩_{1}) \<ra> (x⇩_{1}\<ra>x⇩_{2}\<ra>x⇩_{3}) \<ra> (\<rm>x⇩_{3}) = x⇩_{2}" using cancel_middle(3) by simp qed text‹ We can cancel an element on the right from both sides of an equation. › lemma (in topgroup) cancel_right_add: assumes "x⇩_{1}∈ G" "x⇩_{2}∈ G" "x⇩_{3}∈ G" "x⇩_{1}\<ra>x⇩_{2}= x⇩_{3}\<ra>x⇩_{2}" shows "x⇩_{1}= x⇩_{3}" using assms cancel_right by simp text‹ We can cancel an element on the left from both sides of an equation. › lemma (in topgroup) cancel_left_add: assumes "x⇩_{1}∈ G" "x⇩_{2}∈ G" "x⇩_{3}∈ G" "x⇩_{1}\<ra>x⇩_{2}= x⇩_{1}\<ra>x⇩_{3}" shows "x⇩_{2}= x⇩_{3}" using assms cancel_left by simp text‹We can put an element on the other side of an equation.› lemma (in topgroup) put_on_the_other_side: assumes "x⇩_{1}∈ G" "x⇩_{2}∈ G" "x⇩_{3}= x⇩_{1}\<ra>x⇩_{2}" shows "x⇩_{3}\<ra>(\<rm>x⇩_{2}) = x⇩_{1}" and "(\<rm>x⇩_{1})\<ra>x⇩_{3}= x⇩_{2}" using assms group0_2_L18 by auto text‹A simple equation from lemma ‹simple_equation0› in ‹Group_ZF› in additive notation › lemma (in topgroup) simple_equation0_add: assumes "x⇩_{1}∈ G" "x⇩_{2}∈ G" "x⇩_{3}∈ G" "x⇩_{1}\<ra>(\<rm>x⇩_{2}) = (\<rm>x⇩_{3})" shows "x⇩_{3}= x⇩_{2}\<ra> (\<rm>x⇩_{1})" using assms simple_equation0 by blast text‹A simple equation from lemma ‹simple_equation1› in ‹Group_ZF› in additive notation › lemma (in topgroup) simple_equation1_add: assumes "x⇩_{1}∈ G" "x⇩_{2}∈ G" "x⇩_{3}∈ G" "(\<rm>x⇩_{1})\<ra>x⇩_{2}= (\<rm>x⇩_{3})" shows "x⇩_{3}= (\<rm>x⇩_{2}) \<ra> x⇩_{1}" using assms simple_equation1 by blast text‹The set comprehension form of negative of a set. The proof uses the ‹ginv_image› lemma from ‹Group_ZF› theory which states the same thing in multiplicative notation. › lemma (in topgroup) ginv_image_add: assumes "V⊆G" shows "(\<sm>V)⊆G" and "(\<sm>V) = {\<rm>x. x ∈ V}" using assms ginv_image by auto text‹ The additive notation version of ‹ginv_image_el› lemma from ‹Group_ZF› theory › lemma (in topgroup) ginv_image_el_add: assumes "V⊆G" "x ∈ (\<sm>V)" shows "(\<rm>x) ∈ V" using assms ginv_image_el by simp text‹Of course the product topology is a topology (on $G\times G$).› lemma (in topgroup) prod_top_on_G: shows "τ {is a topology}" and "⋃τ = G×G" using topSpaceAssum Top_1_4_T1 by auto text‹Let's recall that $f$ is a binary operation on $G$ in this context.› lemma (in topgroup) topgroup_f_binop: shows "f : G×G → G" using Ggroup group0_def group0.group_oper_fun by simp text‹A subgroup of a topological group is a topological group with relative topology and restricted operation. Relative topology is the same as ‹T {restricted to} H› which is defined to be $\{V \cap H: V\in T\}$ in ‹ZF1› theory.› lemma (in topgroup) top_subgroup: assumes A1: "IsAsubgroup(H,f)" shows "IsAtopologicalGroup(T {restricted to} H,restrict(f,H×H))" proof - let ?τ⇩_{0}= "T {restricted to} H" let ?f⇩_{H}= "restrict(f,H×H)" have "⋃?τ⇩_{0}= G ∩ H" using union_restrict by simp also from A1 have "… = H" using group0_3_L2 by blast finally have "⋃?τ⇩_{0}= H" by simp have "?τ⇩_{0}{is a topology}" using Top_1_L4 by simp moreover from A1 ‹⋃?τ⇩_{0}= H› have "IsAgroup(⋃?τ⇩_{0},?f⇩_{H})" using IsAsubgroup_def by simp moreover have "IsContinuous(ProductTopology(?τ⇩_{0},?τ⇩_{0}),?τ⇩_{0},?f⇩_{H})" proof - have "two_top_spaces0(τ, T,f)" using topSpaceAssum prod_top_on_G topgroup_f_binop prod_top_on_G two_top_spaces0_def by simp moreover from A1 have "H ⊆ G" using group0_3_L2 by simp then have "H×H ⊆ ⋃τ" using prod_top_on_G by auto moreover have "IsContinuous(τ,T,f)" using fcon by simp ultimately have "IsContinuous(τ {restricted to} H×H, T {restricted to} ?f⇩_{H}``(H×H),?f⇩_{H})" using two_top_spaces0.restr_restr_image_cont by simp moreover have "ProductTopology(?τ⇩_{0},?τ⇩_{0}) = τ {restricted to} H×H" using topSpaceAssum prod_top_restr_comm by simp moreover from A1 have "?f⇩_{H}``(H×H) = H" using image_subgr_op by simp ultimately show ?thesis by simp qed moreover have "IsContinuous(?τ⇩_{0},?τ⇩_{0},GroupInv(⋃?τ⇩_{0},?f⇩_{H}))" proof - let ?g = "restrict(GroupInv(G,f),H)" have "GroupInv(G,f) : G → G" using Ggroup group0_2_T2 by simp then have "two_top_spaces0(T,T,GroupInv(G,f))" using topSpaceAssum two_top_spaces0_def by simp moreover from A1 have "H ⊆ ⋃T" using group0_3_L2 by simp ultimately have "IsContinuous(?τ⇩_{0},T {restricted to} ?g``(H),?g)" using inv_cont two_top_spaces0.restr_restr_image_cont by simp moreover from A1 have "?g``(H) = H" using restr_inv_onto by simp moreover from A1 have "GroupInv(H,?f⇩_{H}) = ?g" using group0_3_T1 by simp with ‹⋃?τ⇩_{0}= H› have "?g = GroupInv(⋃?τ⇩_{0},?f⇩_{H})" by simp ultimately show ?thesis by simp qed ultimately show ?thesis unfolding IsAtopologicalGroup_def by simp qed subsection‹Interval arithmetic, translations and inverse of set› text‹In this section we list some properties of operations of translating a set and reflecting it around the neutral element of the group. Many of the results are proven in other theories, here we just collect them and rewrite in notation specific to the ‹topgroup› context.› text‹Different ways of looking at adding sets.› lemma (in topgroup) interval_add: assumes "A⊆G" "B⊆G" shows "A\<sad>B ⊆ G" "A\<sad>B = f``(A×B)" "A\<sad>B = (⋃x∈A. x\<ltr>B)" "A\<sad>B = {x\<ra>y. ⟨x,y⟩ ∈ A×B}" proof - from assms show "A\<sad>B ⊆ G" and "A\<sad>B = f``(A×B)" and "A\<sad>B = {x\<ra>y. ⟨x,y⟩ ∈ A×B}" using topgroup_f_binop lift_subsets_explained by auto from assms show "A\<sad>B = (⋃x∈A. x\<ltr>B)" using image_ltrans_union by simp qed text‹ If the neutral element is in a set, then it is in the sum of the sets. › lemma (in topgroup) interval_add_zero: assumes "A⊆G" "𝟬∈A" shows "𝟬 ∈ A\<sad>A" proof - from assms have "𝟬\<ra>𝟬 ∈ A\<sad>A" using interval_add(4) by auto then show "𝟬 ∈ A\<sad>A" using group0_2_L2 by auto qed text‹Some lemmas from ‹Group_ZF_1› about images of set by translations written in additive notation› lemma (in topgroup) lrtrans_image: assumes "V⊆G" "x∈G" shows "x\<ltr>V = {x\<ra>v. v∈V}" "V\<rtr>x = {v\<ra>x. v∈V}" using assms ltrans_image rtrans_image by auto text‹ Right and left translations of a set are subsets of the group. This is of course typically applied to the subsets of the group, but formally we don't need to assume that. › lemma (in topgroup) lrtrans_in_group_add: assumes "x∈G" shows "x\<ltr>V ⊆ G" and "V\<rtr>x ⊆G" using assms lrtrans_in_group by auto text‹ A corollary from ‹interval_add› › corollary (in topgroup) elements_in_set_sum: assumes "A⊆G" "B⊆G" "t ∈ A\<sad>B" shows "∃s∈A. ∃q∈B. t=s\<ra>q" using assms interval_add(4) by auto text‹ A corollary from ‹ lrtrans_image› › corollary (in topgroup) elements_in_ltrans: assumes "B⊆G" "g∈G" "t ∈ g\<ltr>B" shows "∃q∈B. t=g\<ra>q" using assms lrtrans_image(1) by simp text‹ Another corollary of ‹lrtrans_image› › corollary (in topgroup) elements_in_rtrans: assumes "B⊆G" "g∈G" "t ∈ B\<rtr>g" shows "∃q∈B. t=q\<ra>g" using assms lrtrans_image(2) by simp text‹Another corollary from ‹interval_add› › corollary (in topgroup) elements_in_set_sum_inv: assumes "A⊆G" "B⊆G" "t=s\<ra>q" "s∈A" "q∈B" shows "t ∈ A\<sad>B" using assms interval_add by auto text‹Another corollary of ‹lrtrans_image› › corollary (in topgroup) elements_in_ltrans_inv: assumes "B⊆G" "g∈G" "q∈B" "t=g\<ra>q" shows "t ∈ g\<ltr>B" using assms lrtrans_image(1) by auto text‹Another corollary of ‹rtrans_image_add› › lemma (in topgroup) elements_in_rtrans_inv: assumes "B⊆G" "g∈G" "q∈B" "t=q\<ra>g" shows "t ∈ B\<rtr>g" using assms lrtrans_image(2) by auto text‹Right and left translations are continuous.› lemma (in topgroup) trans_cont: assumes "g∈G" shows "IsContinuous(T,T,RightTranslation(G,f,g))" and "IsContinuous(T,T,LeftTranslation(G,f,g))" using assms trans_eq_section topgroup_f_binop fcon prod_top_spaces0_valid prod_top_spaces0.fix_1st_var_cont prod_top_spaces0.fix_2nd_var_cont by auto text‹Left and right translations of an open set are open.› lemma (in topgroup) open_tr_open: assumes "g∈G" and "V∈T" shows "g\<ltr>V ∈ T" and "V\<rtr>g ∈ T" using assms neg_in_tgroup trans_cont IsContinuous_def trans_image_vimage by auto text‹Right and left translations are homeomorphisms.› lemma (in topgroup) tr_homeo: assumes "g∈G" shows "IsAhomeomorphism(T,T,RightTranslation(G,f,g))" and "IsAhomeomorphism(T,T,LeftTranslation(G,f,g))" using assms trans_bij trans_cont open_tr_open bij_cont_open_homeo by auto text‹Left translations preserve interior.› lemma (in topgroup) ltrans_interior: assumes A1: "g∈G" and A2: "A⊆G" shows "g \<ltr> int(A) = int(g\<ltr>A)" proof - from assms have "A ⊆ ⋃T" and "IsAhomeomorphism(T,T,LeftTranslation(G,f,g))" using tr_homeo by auto then show ?thesis using int_top_invariant by simp qed text‹Right translations preserve interior.› lemma (in topgroup) rtrans_interior: assumes A1: "g∈G" and A2: "A⊆G" shows "int(A) \<rtr> g = int(A\<rtr>g)" proof - from assms have "A ⊆ ⋃T" and "IsAhomeomorphism(T,T,RightTranslation(G,f,g))" using tr_homeo by auto then show ?thesis using int_top_invariant by simp qed text‹Translating by an inverse and then by an element cancels out.› lemma (in topgroup) trans_inverse_elem: assumes "g∈G" and "A⊆G" shows "g\<ltr>((\<rm>g)\<ltr>A) = A" using assms neg_in_tgroup trans_comp_image group0_2_L6 trans_neutral image_id_same by simp text‹Inverse of an open set is open.› lemma (in topgroup) open_inv_open: assumes "V∈T" shows "(\<sm>V) ∈ T" using assms inv_image_vimage inv_cont IsContinuous_def by simp text‹Inverse is a homeomorphism.› lemma (in topgroup) inv_homeo: shows "IsAhomeomorphism(T,T,GroupInv(G,f))" using group_inv_bij inv_cont open_inv_open bij_cont_open_homeo by simp text‹Taking negative preserves interior.› lemma (in topgroup) int_inv_inv_int: assumes "A ⊆ G" shows "int(\<sm>A) = \<sm>(int(A))" using assms inv_homeo int_top_invariant by simp subsection‹Neighborhoods of zero› text‹Zero neighborhoods are (not necessarily open) sets whose interior contains the neutral element of the group. In the ‹topgroup› locale the collection of neighboorhoods of zero is denoted ‹𝒩⇩_{0}›. › text‹The whole space is a neighborhood of zero.› lemma (in topgroup) zneigh_not_empty: shows "G ∈ 𝒩⇩_{0}" using topSpaceAssum IsATopology_def Top_2_L3 zero_in_tgroup by simp text‹Any element that belongs to a subset of the group belongs to that subset with the interior of a neighborhood of zero added. › lemma (in topgroup) elem_in_int_sad: assumes "A⊆G" "g∈A" "H ∈ 𝒩⇩_{0}" shows "g ∈ A\<sad>int(H)" proof - from assms(3) have "𝟬 ∈ int(H)" and "int(H) ⊆ G" using Top_2_L2 by auto with assms(1,2) have "g\<ra>𝟬 ∈ A\<sad>int(H)" using elements_in_set_sum_inv by simp with assms(1,2) show ?thesis using group0_2_L2 by auto qed text‹Any element belongs to the interior of any neighboorhood of zero left translated by that element.› lemma (in topgroup) elem_in_int_ltrans: assumes "g∈G" and "H ∈ 𝒩⇩_{0}" shows "g ∈ int(g\<ltr>H)" and "g ∈ int(g\<ltr>H) \<sad> int(H)" proof - from assms(2) have "𝟬 ∈ int(H)" and "int(H) ⊆ G" using Top_2_L2 by auto with assms(1) have "g ∈ g \<ltr> int(H)" using neut_trans_elem by simp with assms show "g ∈ int(g\<ltr>H)" using ltrans_interior by simp from assms(1) have "int(g\<ltr>H) ⊆ G" using lrtrans_in_group_add(1) Top_2_L1 by blast with ‹g ∈ int(g\<ltr>H)› assms(2) show "g ∈ int(g\<ltr>H) \<sad> int(H)" using elem_in_int_sad by simp qed text‹Any element belongs to the interior of any neighboorhood of zero right translated by that element.› lemma (in topgroup) elem_in_int_rtrans: assumes A1: "g∈G" and A2: "H ∈ 𝒩⇩_{0}" shows "g ∈ int(H\<rtr>g)" and "g ∈ int(H\<rtr>g) \<sad> int(H)" proof - from A2 have "𝟬 ∈ int(H)" and "int(H) ⊆ G" using Top_2_L2 by auto with A1 have "g ∈ int(H) \<rtr> g" using neut_trans_elem by simp with assms show "g ∈ int(H\<rtr>g)" using rtrans_interior by simp from assms(1) have "int(H\<rtr>g) ⊆ G" using lrtrans_in_group_add(2) Top_2_L1 by blast with ‹g ∈ int(H\<rtr>g)› assms(2) show "g ∈ int(H\<rtr>g) \<sad> int(H)" using elem_in_int_sad by simp qed text‹Negative of a neighborhood of zero is a neighborhood of zero.› lemma (in topgroup) neg_neigh_neigh: assumes "H ∈ 𝒩⇩_{0}" shows "(\<sm>H) ∈ 𝒩⇩_{0}" proof - from assms have "int(H) ⊆ G" and "𝟬 ∈ int(H)" using Top_2_L1 by auto with assms have "𝟬 ∈ int(\<sm>H)" using neut_inv_neut int_inv_inv_int by simp moreover have "GroupInv(G,f):G→G" using Ggroup group0_2_T2 by simp then have "(\<sm>H) ⊆ G" using func1_1_L6 by simp ultimately show ?thesis by simp qed text‹Left translating an open set by a negative of a point that belongs to it makes it a neighboorhood of zero.› lemma (in topgroup) open_trans_neigh: assumes A1: "U∈T" and "g∈U" shows "(\<rm>g)\<ltr>U ∈ 𝒩⇩_{0}" proof - let ?H = "(\<rm>g)\<ltr>U" from assms have "g∈G" by auto then have "(\<rm>g) ∈ G" using neg_in_tgroup by simp with A1 have "?H∈T" using open_tr_open by simp hence "?H ⊆ G" by auto moreover have "𝟬 ∈ int(?H)" proof - from assms have "U⊆G" and "g∈U" by auto with ‹?H∈T› show "𝟬 ∈ int(?H)" using elem_trans_neut Top_2_L3 by auto qed ultimately show ?thesis by simp qed text‹Right translating an open set by a negative of a point that belongs to it makes it a neighboorhood of zero.› lemma (in topgroup) open_trans_neigh_2: assumes A1: "U∈T" and "g∈U" shows "U\<rtr>(\<rm>g) ∈ 𝒩⇩_{0}" proof - let ?H = "U\<rtr>(\<rm>g)" from assms have "g∈G" by auto then have "(\<rm>g) ∈ G" using neg_in_tgroup by simp with A1 have "?H∈T" using open_tr_open by simp hence "?H ⊆ G" by auto moreover have "𝟬 ∈ int(?H)" proof - from assms have "U⊆G" and "g∈U" by auto with ‹?H∈T› show "𝟬 ∈ int(?H)" using elem_trans_neut Top_2_L3 by auto qed ultimately show ?thesis by simp qed text‹Right and left translating an neighboorhood of zero by a point and its negative makes it back a neighboorhood of zero.› lemma (in topgroup) lrtrans_neigh: assumes "W∈𝒩⇩_{0}" and "x∈G" shows "x\<ltr>(W\<rtr>(\<rm>x)) ∈ 𝒩⇩_{0}" and "(x\<ltr>W)\<rtr>(\<rm>x) ∈ 𝒩⇩_{0}" proof - from assms(2) have "x\<ltr>(W\<rtr>(\<rm>x)) ⊆ G" using lrtrans_in_group_add(1) by simp moreover have "𝟬 ∈ int(x\<ltr>(W\<rtr>(\<rm>x)))" proof - from assms(2) have "int(W\<rtr>(\<rm>x)) ⊆ G" using neg_in_tgroup lrtrans_in_group_add(2) Top_2_L1 by blast with assms(2) have "(x\<ltr>int((W\<rtr>(\<rm>x)))) = {x\<ra>y. y∈int(W\<rtr>(\<rm>x))}" using lrtrans_image(1) by simp moreover from assms have "(\<rm>x) ∈ int(W\<rtr>(\<rm>x))" using neg_in_tgroup elem_in_int_rtrans(1) by simp ultimately have "x\<ra>(\<rm>x) ∈ x\<ltr>int(W\<rtr>(\<rm>x))" by auto with assms show ?thesis using group0_2_L6 neg_in_tgroup lrtrans_in_group_add(2) ltrans_interior by simp qed ultimately show "x\<ltr>(W\<rtr>(\<rm>x)) ∈ 𝒩⇩_{0}" by simp from assms(2) have "(x\<ltr>W)\<rtr>(\<rm>x) ⊆ G" using lrtrans_in_group_add(2) neg_in_tgroup by simp moreover have "𝟬 ∈ int((x\<ltr>W)\<rtr>(\<rm>x))" proof - from assms(2) have "int((x\<ltr>W)) ⊆ G" using lrtrans_in_group_add(1) Top_2_L1 by blast with assms(2) have "int(x\<ltr>W) \<rtr> (\<rm>x) = {y\<ra>(\<rm>x).y∈int(x\<ltr>W)}" using neg_in_tgroup lrtrans_image(2) by simp moreover from assms have "x ∈ int(x\<ltr>W)" using elem_in_int_ltrans(1) by simp ultimately have "x\<ra>(\<rm>x) ∈ int(x\<ltr>W) \<rtr> (\<rm>x)" by auto with assms(2) have "𝟬 ∈ int(x\<ltr>W) \<rtr> (\<rm>x)" using group0_2_L6 by simp with assms show ?thesis using group0_2_L6 neg_in_tgroup lrtrans_in_group_add(1) rtrans_interior by auto qed ultimately show "(x\<ltr>W)\<rtr>(\<rm>x) ∈ 𝒩⇩_{0}" by simp qed text‹If $A$ is a subset of $B$ translated by $-x$ then its translation by $x$ is a subset of $B$.› lemma (in topgroup) trans_subset: assumes "A ⊆ ((\<rm>x)\<ltr>B)""x∈G" "B⊆G" shows "x\<ltr>A ⊆ B" proof- from assms(1) have "x\<ltr>A ⊆ (x\<ltr> ((\<rm>x)\<ltr>B))" by auto with assms(2,3) show "x\<ltr>A ⊆ B" using neg_in_tgroup trans_comp_image group0_2_L6 trans_neutral image_id_same by simp qed text‹ Every neighborhood of zero has a symmetric subset that is a neighborhood of zero.› theorem (in topgroup) exists_sym_zerohood: assumes "U∈𝒩⇩_{0}" shows "∃V∈𝒩⇩_{0}. (V⊆U ∧ (\<sm>V)=V)" proof let ?V = "U∩(\<sm>U)" have "U⊆G" using assms unfolding zerohoods_def by auto then have "?V⊆G" by auto have invg:" GroupInv(G, f) ∈ G → G" using group0_2_T2 Ggroup by auto have invb:"GroupInv(G, f) ∈bij(G,G)" using group_inv_bij(2) by auto have "(\<sm>?V)=GroupInv(G,f)-``?V" unfolding setninv_def using inv_image_vimage by auto also have "…=(GroupInv(G,f)-``U)∩(GroupInv(G,f)-``(\<sm>U))" using invim_inter_inter_invim invg by auto also have "…=(\<sm>U)∩(GroupInv(G,f)-``(GroupInv(G,f)``U))" unfolding setninv_def using inv_image_vimage by auto also from ‹U⊆G› have "…=(\<sm>U)∩U" using inj_vimage_image invb unfolding bij_def by auto finally have "(\<sm>?V)=?V" by auto then show "?V ⊆ U ∧ (\<sm> ?V) = ?V" by auto from assms have "(\<sm>U)∈𝒩⇩_{0}" using neg_neigh_neigh by auto with assms have "𝟬 ∈ int(U)∩int(\<sm>U)" unfolding zerohoods_def by auto moreover have "int(U)∩int(\<sm>U) = int(?V)" using int_inter_int by simp ultimately have "𝟬 ∈ int(?V)" by (rule set_mem_eq) with ‹?V⊆G› show "?V∈𝒩⇩_{0}" using zerohoods_def by auto qed text‹ We can say even more than in ‹exists_sym_zerohood›: every neighborhood of zero $U$ has a symmetric subset that is a neighborhood of zero and its set double is contained in $U$.› theorem (in topgroup) exists_procls_zerohood: assumes "U∈𝒩⇩_{0}" shows "∃V∈𝒩⇩_{0}. (V⊆U∧ (V\<sad>V)⊆U ∧ (\<sm>V)=V)" proof- have "int(U)∈T" using Top_2_L2 by auto then have "f-``(int(U))∈τ" using fcon IsContinuous_def by auto moreover have fne:"f ` ⟨𝟬, 𝟬⟩ = 𝟬" using group0_2_L2 by auto moreover have "𝟬∈int(U)" using assms unfolding zerohoods_def by auto then have "f -`` {𝟬}⊆f-``(int(U))" using func1_1_L8 vimage_def by auto then have "GroupInv(G,f)⊆f-``(int(U))" using group0_2_T3 by auto then have "⟨𝟬,𝟬⟩∈f-``(int(U))" using fne zero_in_tgroup unfolding GroupInv_def by auto ultimately obtain W V where wop:"W∈T" and vop:"V∈T" and cartsub:"W×V⊆f-``(int(U))" and zerhood:"⟨𝟬,𝟬⟩∈W×V" using prod_top_point_neighb topSpaceAssum unfolding prodtop_def by force then have "𝟬∈W" and "𝟬∈V" by auto then have "𝟬∈W∩V" by auto have sub:"W∩V⊆G" using wop vop G_def by auto have assoc:"f∈G×G→G" using group_oper_fun by auto { fix t s assume "t∈W∩V" and "s∈W∩V" then have "t∈W" and "s∈V" by auto then have "⟨t,s⟩∈W×V" by auto then have "⟨t,s⟩∈f-``(int(U))" using cartsub by auto then have "f`⟨t,s⟩∈int(U)" using func1_1_L15 assoc by auto } hence "{f`⟨t,s⟩. ⟨t,s⟩∈(W∩V)×(W∩V)}⊆int(U)" by auto then have "(W∩V)\<sad>(W∩V)⊆int(U)" unfolding setadd_def using lift_subsets_explained(4) assoc sub by auto then have "(W∩V)\<sad>(W∩V)⊆U" using Top_2_L1 by auto from topSpaceAssum have "W∩V∈T" using vop wop unfolding IsATopology_def by auto then have "int(W∩V)=W∩V" using Top_2_L3 by auto with sub ‹𝟬∈W∩V› have "W∩V∈𝒩⇩_{0}" unfolding zerohoods_def by auto then obtain Q where "Q∈𝒩⇩_{0}" and "Q⊆W∩V" and "(\<sm>Q)=Q" using exists_sym_zerohood by blast then have "Q×Q⊆(W∩V)×(W∩V)" by auto moreover from ‹Q⊆W∩V› have "W∩V⊆G" and "Q⊆G" using vop wop unfolding G_def by auto ultimately have "Q\<sad>Q⊆(W∩V)\<sad>(W∩V)" using interval_add(2) func1_1_L8 by auto with ‹(W∩V)\<sad>(W∩V)⊆U› have "Q\<sad>Q⊆U" by auto from ‹Q∈𝒩⇩_{0}› have "𝟬∈Q" unfolding zerohoods_def using Top_2_L1 by auto with ‹Q\<sad>Q⊆U› ‹Q⊆G› have "𝟬\<ltr>Q⊆U" using interval_add(3) by auto with ‹Q⊆G› have "Q⊆U" unfolding ltrans_def gzero_def using trans_neutral(2) image_id_same by auto with ‹Q∈𝒩⇩_{0}› ‹Q\<sad>Q⊆U› ‹(\<sm>Q)=Q› show ?thesis by auto qed subsection‹Closure in topological groups› text‹This section is devoted to a characterization of closure in topological groups.› text‹Closure of a set is contained in the sum of the set and any neighboorhood of zero.› lemma (in topgroup) cl_contains_zneigh: assumes A1: "A⊆G" and A2: "H ∈ 𝒩⇩_{0}" shows "cl(A) ⊆ A\<sad>H" proof fix x assume "x ∈ cl(A)" from A1 have "cl(A) ⊆ G" using Top_3_L11 by simp with ‹x ∈ cl(A)› have "x∈G" by auto have "int(H) ⊆ G" using Top_2_L2 by auto let ?V = "int(x \<ltr> (\<sm>H))" have "?V = x \<ltr> (\<sm>int(H))" proof - from A2 ‹x∈G› have "?V = x \<ltr> int(\<sm>H)" using neg_neigh_neigh ltrans_interior by simp with A2 show ?thesis using int_inv_inv_int by simp qed have "A∩?V ≠ 0" proof - from A2 ‹x∈G› ‹x ∈ cl(A)› have "?V∈T" and "x ∈ cl(A) ∩ ?V" using neg_neigh_neigh elem_in_int_ltrans(1) Top_2_L2 by auto with A1 show "A∩?V ≠ 0" using cl_inter_neigh by simp qed then obtain y where "y∈A" and "y∈?V" by auto with ‹?V = x \<ltr> (\<sm>int(H))› ‹int(H) ⊆ G› ‹x∈G› have "x ∈ y\<ltr>int(H)" using ltrans_inv_in by simp with ‹y∈A› have "x ∈ (⋃y∈A. y\<ltr>H)" using Top_2_L1 func1_1_L8 by auto with assms show "x ∈ A\<sad>H" using interval_add(3) by simp qed text‹The next theorem provides a characterization of closure in topological groups in terms of neighborhoods of zero.› theorem (in topgroup) cl_topgroup: assumes "A⊆G" shows "cl(A) = (⋂H∈𝒩⇩_{0}. A\<sad>H)" proof from assms show "cl(A) ⊆ (⋂H∈𝒩⇩_{0}. A\<sad>H)" using zneigh_not_empty cl_contains_zneigh by auto next { fix x assume "x ∈ (⋂H∈𝒩⇩_{0}. A\<sad>H)" then have "x ∈ A\<sad>G" using zneigh_not_empty by auto with assms have "x∈G" using interval_add by blast have "∀U∈T. x∈U ⟶ U∩A ≠ 0" proof - { fix U assume "U∈T" and "x∈U" let ?H = "\<sm>((\<rm>x)\<ltr>U)" from ‹U∈T› and ‹x∈U› have "(\<rm>x)\<ltr>U ⊆ G" and "?H ∈ 𝒩⇩_{0}" using open_trans_neigh neg_neigh_neigh by auto with ‹x ∈ (⋂H∈𝒩⇩_{0}. A\<sad>H)› have "x ∈ A\<sad>?H" by auto with assms ‹?H ∈ 𝒩⇩_{0}› obtain y where "y∈A" and "x ∈ y\<ltr>?H" using interval_add(3) by auto have "y∈U" proof - from assms ‹y∈A› have "y∈G" by auto with ‹(\<rm>x)\<ltr>U ⊆ G› and ‹x ∈ y\<ltr>?H› have "y ∈ x\<ltr>((\<rm>x)\<ltr>U)" using ltrans_inv_in by simp with ‹U∈T› ‹x∈G› show "y∈U" using neg_in_tgroup trans_comp_image group0_2_L6 trans_neutral image_id_same by auto qed with ‹y∈A› have "U∩A ≠ 0" by auto } thus ?thesis by simp qed with assms ‹x∈G› have "x ∈ cl(A)" using inter_neigh_cl by simp } thus "(⋂H∈𝒩⇩_{0}. A\<sad>H) ⊆ cl(A)" by auto qed subsection‹Sums of sequences of elements and subsets› text‹In this section we consider properties of the function $G^n\rightarrow G$, $x=(x_0,x_1,...,x_{n-1})\mapsto \sum_{i=0}^{n-1}x_i$. We will model the cartesian product $G^n$ by the space of sequences $n\rightarrow G$, where $n=\{0,1,...,n-1 \}$ is a natural number. This space is equipped with a natural product topology defined in ‹Topology_ZF_3›.› text‹Let's recall first that the sum of elements of a group is an element of the group.› lemma (in topgroup) sum_list_in_group: assumes "n ∈ nat" and "x: succ(n)→G" shows "(∑x) ∈ G" proof - from assms have "semigr0(G,f)" and "n ∈ nat" "x: succ(n)→G" using semigr0_valid_in_tgroup by auto then have "Fold1(f,x) ∈ G" by (rule semigr0.prod_type) thus "(∑x) ∈ G" by simp qed text‹In this context ‹x\<ra>y› is the same as the value of the group operation on the elements $x$ and $y$. Normally we shouldn't need to state this a s separate lemma.› lemma (in topgroup) grop_def1: shows "f`⟨x,y⟩ = x\<ra>y" by simp text‹Another theorem from ‹Semigroup_ZF› theory that is useful to have in the additive notation.› lemma (in topgroup) shorter_set_add: assumes "n ∈ nat" and "x: succ(succ(n))→G" shows "(∑x) = (∑Init(x)) \<ra> (x`(succ(n)))" proof - from assms have "semigr0(G,f)" and "n ∈ nat" "x: succ(succ(n))→G" using semigr0_valid_in_tgroup by auto then have "Fold1(f,x) = f`⟨Fold1(f,Init(x)),x`(succ(n))⟩" by (rule semigr0.shorter_seq) thus ?thesis by simp qed text‹Sum is a continuous function in the product topology.› theorem (in topgroup) sum_continuous: assumes "n ∈ nat" shows "IsContinuous(SeqProductTopology(succ(n),T),T,{⟨x,∑x⟩.x∈succ(n)→G})" proof - note ‹n ∈ nat› moreover have "IsContinuous(SeqProductTopology(succ(0),T),T,{⟨x,∑x⟩.x∈succ(0)→G})" proof - have "{⟨x,∑x⟩.x∈succ(0)→G} = {⟨x,x`(0)⟩. x∈1→G}" using semigr0_valid_in_tgroup semigr0.prod_of_1elem by simp moreover have "IsAhomeomorphism(SeqProductTopology(1,T),T,{⟨x,x`(0)⟩. x∈1→⋃T})" using topSpaceAssum singleton_prod_top1 by simp ultimately show ?thesis using IsAhomeomorphism_def by simp qed moreover have "∀k∈nat. IsContinuous(SeqProductTopology(succ(k),T),T,{⟨x,∑x⟩.x∈succ(k)→G}) ⟶ IsContinuous(SeqProductTopology(succ(succ(k)),T),T,{⟨x,∑x⟩.x∈succ(succ(k))→G})" proof - { fix k assume "k ∈ nat" let ?s = "{⟨x,∑x⟩.x∈succ(k)→G}" let ?g = "{⟨p,⟨?s`(fst(p)),snd(p)⟩⟩. p ∈ (succ(k)→G)×G}" let ?h = "{⟨x,⟨Init(x),x`(succ(k))⟩⟩. x ∈ succ(succ(k))→G}" let ?φ = "SeqProductTopology(succ(k),T)" let ?ψ = "SeqProductTopology(succ(succ(k)),T)" assume "IsContinuous(?φ,T,?s)" from ‹k ∈ nat› have "?s: (succ(k)→G) → G" using sum_list_in_group ZF_fun_from_total by simp have "?h: (succ(succ(k))→G)→(succ(k)→G)×G" proof - { fix x assume "x ∈ succ(succ(k))→G" with ‹k ∈ nat› have "Init(x) ∈ (succ(k)→G)" using init_props by simp with ‹k ∈ nat› ‹x : succ(succ(k))→G› have "⟨Init(x),x`(succ(k))⟩ ∈ (succ(k)→G)×G" using apply_funtype by blast } then show ?thesis using ZF_fun_from_total by simp qed moreover have "?g:((succ(k)→G)×G)→(G×G)" proof - { fix p assume "p ∈ (succ(k)→G)×G" hence "fst(p): succ(k)→G" and "snd(p) ∈ G" by auto with ‹?s: (succ(k)→G) → G› have "⟨?s`(fst(p)),snd(p)⟩ ∈ G×G" using apply_funtype by blast } then show "?g:((succ(k)→G)×G)→(G×G)" using ZF_fun_from_total by simp qed moreover have "f : G×G → G" using topgroup_f_binop by simp ultimately have "f O ?g O ?h :(succ(succ(k))→G)→G" using comp_fun by blast from ‹k ∈ nat› have "IsContinuous(?ψ,ProductTopology(?φ,T),?h)" using topSpaceAssum finite_top_prod_homeo IsAhomeomorphism_def by simp moreover have "IsContinuous(ProductTopology(?φ,T),τ,?g)" proof - from topSpaceAssum have "T {is a topology}" "?φ {is a topology}" "⋃?φ = succ(k)→G" using seq_prod_top_is_top by auto moreover from ‹⋃?φ = succ(k)→G› ‹?s: (succ(k)→G) → G› have "?s:⋃?φ→⋃T" by simp moreover note ‹IsContinuous(?φ,T,?s)› moreover from ‹⋃?φ = succ(k)→G› have "?g = {⟨p,⟨?s`(fst(p)),snd(p)⟩⟩. p ∈ ⋃?φ×⋃T}" by simp ultimately have "IsContinuous(ProductTopology(?φ,T),ProductTopology(T,T),?g)" using cart_prod_cont1 by blast thus ?thesis by simp qed moreover have "IsContinuous(τ,T,f)" using fcon by simp moreover have "{⟨x,∑x⟩.x∈succ(succ(k))→G} = f O ?g O ?h" proof - let ?d = "{⟨x,∑x⟩.x∈succ(succ(k))→G}" from ‹k∈nat› have "∀x∈succ(succ(k))→G. (∑x) ∈ G" using sum_list_in_group by blast then have "?d:(succ(succ(k))→G)→G" using sum_list_in_group ZF_fun_from_total by simp moreover note ‹f O ?g O ?h :(succ(succ(k))→G)→G› moreover have "∀x∈succ(succ(k))→G. ?d`(x) = (f O ?g O ?h)`(x)" proof fix x assume "x∈succ(succ(k))→G" then have I: "?h`(x) = ⟨Init(x),x`(succ(k))⟩" using ZF_fun_from_tot_val1 by simp moreover from ‹k∈nat› ‹x∈succ(succ(k))→G› have "Init(x): succ(k)→G" using init_props by simp moreover from ‹k∈nat› ‹x:succ(succ(k))→G› have II: "x`(succ(k)) ∈ G" using apply_funtype by blast ultimately have "?h`(x) ∈ (succ(k)→G)×G" by simp then have "?g`(?h`(x)) = ⟨?s`(fst(?h`(x))),snd(?h`(x))⟩" using ZF_fun_from_tot_val1 by simp with I have "?g`(?h`(x)) = ⟨?s`(Init(x)),x`(succ(k))⟩" by simp with ‹Init(x): succ(k)→G› have "?g`(?h`(x)) = ⟨∑Init(x),x`(succ(k))⟩" using ZF_fun_from_tot_val1 by simp with ‹k ∈ nat› ‹x: succ(succ(k))→G› have "f`(?g`(?h`(x))) = (∑x)" using shorter_set_add by simp with ‹x ∈ succ(succ(k))→G› have "f`(?g`(?h`(x))) = ?d`(x)" using ZF_fun_from_tot_val1 by simp moreover from ‹?h: (succ(succ(k))→G)→(succ(k)→G)×G› ‹?g:((succ(k)→G)×G)→(G×G)› ‹f:(G×G)→G› ‹x∈succ(succ(k))→G› have "(f O ?g O ?h)`(x) = f`(?g`(?h`(x)))" by (rule func1_1_L18) ultimately show "?d`(x) = (f O ?g O ?h)`(x)" by simp qed ultimately show "{⟨x,∑x⟩.x∈succ(succ(k))→G} = f O ?g O ?h" using func_eq by simp qed moreover note ‹IsContinuous(τ,T,f)› ultimately have "IsContinuous(?ψ,T,{⟨x,∑x⟩.x∈succ(succ(k))→G})" using comp_cont3 by simp } thus ?thesis by simp qed ultimately show ?thesis by (rule ind_on_nat) qed end