P. 194 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cntzssv 19301	The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑍‘𝑆) ⊆ 𝐵
Theorem	cntzi 19302	Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	elcntr 19303*	Elementhood in the center of a magma. (Contributed by SN, 21-Mar-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
Theorem	cntrss 19304	The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (Cntr‘𝑀) ⊆ 𝐵
Theorem	cntri 19305	Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	resscntz 19306	Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝑌 = (Cntz‘𝐻)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ⊆ 𝐴) → (𝑌‘𝑆) = ((𝑍‘𝑆) ∩ 𝐴))
Theorem	cntzsgrpcl 19307*	Centralizers are closed under the semigroup operation. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    &   ⊢ 𝐶 = (𝑍‘𝑆)    ⇒   ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)
Theorem	cntz2ss 19308	Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇))
Theorem	cntzrec 19309	Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑇 ⊆ (𝑍‘𝑆)))
Theorem	cntziinsn 19310*	Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})))
Theorem	cntzsubm 19311	Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
Theorem	cntzsubg 19312	Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑀))
Theorem	cntzidss 19313	If the elements of 𝑆 commute, the elements of a subset 𝑇 also commute. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ (𝑍‘𝑇))
Theorem	cntzmhm 19314	Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝑌 = (Cntz‘𝐻)    ⇒   ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)))
Theorem	cntzmhm2 19315	Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝑌 = (Cntz‘𝐻)    ⇒   ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)))
Theorem	cntrsubgnsg 19316	A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))
Theorem	cntrnsg 19317	The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ (𝑀 ∈ Grp → 𝑍 ∈ (NrmSGrp‘𝑀))
10.2.9  The opposite group
Syntax	coppg 19318	The opposite group operation.
class oppg
Definition	df-oppg 19319	Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 20315 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ oppg = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), tpos (+g‘𝑤)⟩))
Theorem	oppgval 19320	Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
⊢ + = (+g‘𝑅)    &   ⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
Theorem	oppgplusfval 19321	Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
⊢ + = (+g‘𝑅)    &   ⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ✚ = (+g‘𝑂)    ⇒   ⊢ ✚ = tpos +
Theorem	oppgplus 19322	Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
⊢ + = (+g‘𝑅)    &   ⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ✚ = (+g‘𝑂)    ⇒   ⊢ (𝑋 ✚ 𝑌) = (𝑌 + 𝑋)
Theorem	setsplusg 19323	The other components of an extensible structure remain unchanged if the +g component is set/substituted. (Contributed by Stefan O'Rear, 26-Aug-2015.) Generalisation of the former oppglem and mgplem. (Revised by AV, 18-Oct-2024.)
⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), 𝑆⟩)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (+g‘ndx)    ⇒   ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)
Theorem	oppgbas 19324	Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ 𝐵 = (Base‘𝑂)
Theorem	oppgtset 19325	Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopSet‘𝑅)    ⇒   ⊢ 𝐽 = (TopSet‘𝑂)
Theorem	oppgtopn 19326	Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ 𝐽 = (TopOpen‘𝑂)
Theorem	oppgmnd 19327	The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd)
Theorem	oppgmndb 19328	Bidirectional form of oppgmnd 19327. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
Theorem	oppgid 19329	Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ 0 = (0g‘𝑂)
Theorem	oppggrp 19330	The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp)
Theorem	oppggrpb 19331	Bidirectional form of oppggrp 19330. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
Theorem	oppginv 19332	Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐼 = (invg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂))
Theorem	invoppggim 19333	The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂))
Theorem	oppggic 19334	Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐺 ≃𝑔 𝑂)
Theorem	oppgsubm 19335	Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂)
Theorem	oppgsubg 19336	Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂)
Theorem	oppgcntz 19337	A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝑂 = (oppg‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ (𝑍‘𝐴) = ((Cntz‘𝑂)‘𝐴)
Theorem	oppgcntr 19338	The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝑂 = (oppg‘𝐺)    &   ⊢ 𝑍 = (Cntr‘𝐺)    ⇒   ⊢ 𝑍 = (Cntr‘𝑂)
Theorem	gsumwrev 19339	A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑂 = (oppg‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊)))
Theorem	oppgle 19340	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    ⇒   ⊢ ≤ = (le‘𝑂)
Theorem	oppglt 19341	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ < = (lt‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → < = (lt‘𝑂))
10.2.10  Symmetric groups
10.2.10.1  Definition and basic properties According to Wikipedia ("Symmetric group", 09-Mar-2019, https://en.wikipedia.org/wiki/symmetric_group) "In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions." and according to Encyclopedia of Mathematics ("Symmetric group", 09-Mar-2019, https://www.encyclopediaofmath.org/index.php/Symmetric_group) "The group of all permutations (self-bijections) of a set with the operation of composition (see Permutation group).". In [Rotman] p. 27 "If X is a nonempty set, a permutation of X is a function a : X -> X that is a one-to-one correspondence." and "If X is a nonempty set, the symmetric group on X, denoted SX, is the group whose elements are the permutations of X and whose binary operation is composition of functions.". Therefore, we define the symmetric group on a set 𝐴 as the set of one-to-one onto functions from 𝐴 to itself under function composition, see df-symg 19343. However, the set is allowed to be empty, see symgbas0 19362. Hint: The symmetric groups should not be confused with "symmetry groups" which is a different topic in group theory. In this context, the one-to-one onto functions are called permutations for short. Since the base set of symmetric groups on a set 𝐴 is the set of all permutations of 𝐴 (see symgbas 19345), we can formally say 𝑃 ∈ (SymGrp‘𝐴) expressing "𝑃 is a permutation of 𝐴" if we are not interested in the group (or topology) structure. In general, a permutation group "... is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself)." (see Wikipedia "Permutation group", 17-Mar-2019, https://en.wikipedia.org/wiki/Permutation_group 19345). This means that a symmetric group is a permutation group, and each permutation group is a subgroup of a symmetric group (see pgrpsubgsymgbi 19381 and pgrpsubgsymg 19382). For example, the structure with the singleton containing only the identity function restricted to a set as base set and the function composition as group operation is a permutation group (group consisting of permutations), see idrespermg 19384, which is a (proper) subgroup of a symmetric group, see idressubgsymg 19383. As in [Rotman] p. 28 "Let 𝑥 ∈ 𝑋 and 𝑝 ∈ SymGrp(𝑋); we say 𝑝 fixes 𝑥 if (𝑝‘𝑥) = 𝑥; otherwise 𝑝 moves 𝑥.". The theorems starting with symgfix2 19389 are about fixed/moved elements.
Syntax	csymg 19342	Extend class notation to include the class of symmetric groups.
class SymGrp
Definition	df-symg 19343*	Define the symmetric group on set 𝑥. We represent the group as the set of one-to-one onto functions from 𝑥 to itself under function composition, and topologize it as a function space assuming the set is discrete. This definition is based on the fact that a symmetric group is a restriction of the monoid of endofunctions. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 28-Mar-2024.)
⊢ SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))
Theorem	symgval 19344*	The value of the symmetric group function at 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 28-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}    ⇒   ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
Theorem	symgbas 19345*	The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.) (Proof shortened by AV, 29-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
Theorem	elsymgbas2 19346	Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Mario Carneiro, 28-Jan-2015.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴))
Theorem	elsymgbas 19347	Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴))
Theorem	symgbasf1o 19348	Elements in the symmetric group are 1-1 onto functions. (Contributed by SO, 9-Jul-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴)
Theorem	symgbasf 19349	A permutation (element of the symmetric group) is a function from a set into itself. (Contributed by AV, 1-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴⟶𝐴)
Theorem	symgbasmap 19350	A permutation (element of the symmetric group) is a mapping (or set exponentiation) from a set into itself. (Contributed by AV, 30-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (𝐴 ↑m 𝐴))
Theorem	symghash 19351	The symmetric group on 𝑛 objects has cardinality 𝑛!. (Contributed by Mario Carneiro, 22-Jan-2015.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ Fin → (♯‘𝐵) = (!‘(♯‘𝐴)))
Theorem	symgbasfi 19352	The symmetric group on a finite index set is finite. (Contributed by SO, 9-Jul-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ Fin → 𝐵 ∈ Fin)
Theorem	symgfv 19353	The function value of a permutation. (Contributed by AV, 1-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐴)
Theorem	symgfvne 19354	The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍)))
Theorem	symgressbas 19355	The symmetric group on 𝐴 characterized as structure restriction of the monoid of endofunctions on 𝐴 to its base set. (Contributed by AV, 30-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑀 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 = (𝑀 ↾s 𝐵)
Theorem	symgplusg 19356*	The group operation of a symmetric group is the function composition. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) (Proof shortened by AV, 19-Feb-2024.) (Revised by AV, 29-Mar-2024.) (Proof shortened by AV, 14-Aug-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (𝐴 ↑m 𝐴)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
Theorem	symgov 19357	The value of the group operation of the symmetric group on 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) (Revised by AV, 30-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌))
Theorem	symgcl 19358	The group operation of the symmetric group on 𝐴 is closed, i.e. a magma. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by Mario Carneiro, 28-Jan-2015.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	idresperm 19359	The identity function restricted to a set is a permutation of this set. (Contributed by AV, 17-Mar-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))
Theorem	symgmov1 19360*	For a permutation of a set, each element of the set replaces an(other) element of the set. (Contributed by AV, 2-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ (𝑄 ∈ 𝑃 → ∀𝑛 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑛) = 𝑘)
Theorem	symgmov2 19361*	For a permutation of a set, each element of the set is replaced by an(other) element of the set. (Contributed by AV, 2-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ (𝑄 ∈ 𝑃 → ∀𝑛 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑛)
Theorem	symgbas0 19362	The base set of the symmetric group on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Feb-2019.)
⊢ (Base‘(SymGrp‘∅)) = {∅}
Theorem	symg1hash 19363	The symmetric group on a singleton has cardinality 1. (Contributed by AV, 9-Dec-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (♯‘𝐵) = 1)
Theorem	symg1bas 19364	The symmetric group on a singleton is the symmetric group S1 consisting of the identity only. (Contributed by AV, 9-Dec-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{⟨𝐼, 𝐼⟩}})
Theorem	symg2hash 19365	The symmetric group on a (proper) pair has cardinality 2. (Contributed by AV, 9-Dec-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼, 𝐽}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐵) = 2)
Theorem	symg2bas 19366	The symmetric group on a pair is the symmetric group S2 consisting of the identity and the transposition. Notice that this statement is valid for proper pairs only. In the case that both elements are identical, i.e., the pairs are actually singletons, this theorem would be about S1, see Theorem symg1bas 19364. (Contributed by AV, 9-Dec-2018.) (Proof shortened by AV, 16-Jun-2022.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼, 𝐽}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊) → 𝐵 = {{⟨𝐼, 𝐼⟩, ⟨𝐽, 𝐽⟩}, {⟨𝐼, 𝐽⟩, ⟨𝐽, 𝐼⟩}})
Theorem	0symgefmndeq 19367	The symmetric group on the empty set is identical with the monoid of endofunctions on the empty set. (Contributed by AV, 30-Mar-2024.)
⊢ (EndoFMnd‘∅) = (SymGrp‘∅)
Theorem	snsymgefmndeq 19368	The symmetric group on a singleton 𝐴 is identical with the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Mar-2024.)
⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
Theorem	symgpssefmnd 19369	For a set 𝐴 with more than one element, the symmetric group on 𝐴 is a proper subset of the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Mar-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    &   ⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘𝑀))
Theorem	symgvalstruct 19370*	The value of the symmetric group function at 𝐴 represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.) (Proof shortened by AV, 6-Nov-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}    &   ⊢ 𝑀 = (𝐴 ↑m 𝐴)    &   ⊢ + = (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔))    &   ⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Theorem	symgsubmefmnd 19371	The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 18-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    &   ⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐵 ∈ (SubMnd‘𝑀))
Theorem	symgtset 19372	The topology of the symmetric group on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just bijections - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof revised by AV, 30-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))
Theorem	symggrp 19373	The symmetric group on a set 𝐴 is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 28-Jan-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp)
Theorem	symgid 19374	The group identity element of the symmetric group on a set 𝐴. (Contributed by Paul Chapman, 25-Jul-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 1-Apr-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝐺))
Theorem	symginv 19375	The group inverse in the symmetric group corresponds to the functional inverse. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ (𝐹 ∈ 𝐵 → (𝑁‘𝐹) = ◡𝐹)
Theorem	symgsubmefmndALT 19376	The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. Alternate proof based on issubmndb 18771 and not on injsubmefmnd 18863 and sursubmefmnd 18862. (Contributed by AV, 18-Feb-2024.) (Revised by AV, 30-Mar-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    &   ⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐵 ∈ (SubMnd‘𝑀))
Theorem	galactghm 19377*	The currying of a group action is a group homomorphism between the group 𝐺 and the symmetric group (SymGrp‘𝑌). (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐻 = (SymGrp‘𝑌)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)))    ⇒   ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
Theorem	lactghmga 19378*	The converse of galactghm 19377. The uncurrying of a homomorphism into (SymGrp‘𝑌) is a group action. Thus, group actions and group homomorphisms into a symmetric group are essentially equivalent notions. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐻 = (SymGrp‘𝑌)    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝐹‘𝑥)‘𝑦))    ⇒   ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ⊕ ∈ (𝐺 GrpAct 𝑌))
Theorem	symgtopn 19379	The topology of the symmetric group on 𝐴. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺))
Theorem	symgga 19380*	The symmetric group induces a group action on its base set. (Contributed by Mario Carneiro, 24-Jan-2015.)
⊢ 𝐺 = (SymGrp‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = (𝑓 ∈ 𝐵, 𝑥 ∈ 𝑋 ↦ (𝑓‘𝑥))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐹 ∈ (𝐺 GrpAct 𝑋))
Theorem	pgrpsubgsymgbi 19381	Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝑃 ∈ (SubGrp‘𝐺) ↔ (𝑃 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑃) ∈ Grp)))
Theorem	pgrpsubgsymg 19382*	Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.) (Revised by AV, 30-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = (Base‘𝑃)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ((𝑃 ∈ Grp ∧ 𝐹 ⊆ 𝐵 ∧ (+g‘𝑃) = (𝑓 ∈ 𝐹, 𝑔 ∈ 𝐹 ↦ (𝑓 ∘ 𝑔))) → 𝐹 ∈ (SubGrp‘𝐺)))
Theorem	idressubgsymg 19383	The singleton containing only the identity function restricted to a set is a subgroup of the symmetric group of this set. (Contributed by AV, 17-Mar-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {( I ↾ 𝐴)} ∈ (SubGrp‘𝐺))
Theorem	idrespermg 19384	The structure with the singleton containing only the identity function restricted to a set as base set and the function composition as group operation (constructed by (structure) restricting the symmetric group to that singleton) is a permutation group (group consisting of permutations). (Contributed by AV, 17-Mar-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐸 = (𝐺 ↾s {( I ↾ 𝐴)})    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐸 ∈ Grp ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))
10.2.10.2  Cayley's theorem
Theorem	cayleylem1 19385*	Lemma for cayley 19387. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐻 = (SymGrp‘𝑋)    &   ⊢ 𝑆 = (Base‘𝐻)    &   ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))    ⇒   ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻))
Theorem	cayleylem2 19386*	Lemma for cayley 19387. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐻 = (SymGrp‘𝑋)    &   ⊢ 𝑆 = (Base‘𝐻)    &   ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))    ⇒   ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→𝑆)
Theorem	cayley 19387*	Cayley's Theorem (constructive version): given group 𝐺, 𝐹 is an isomorphism between 𝐺 and the subgroup 𝑆 of the symmetric group 𝐻 on the underlying set 𝑋 of 𝐺. See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐻 = (SymGrp‘𝑋)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))    &   ⊢ 𝑆 = ran 𝐹    ⇒   ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐻) ∧ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)) ∧ 𝐹:𝑋–1-1-onto→𝑆))
Theorem	cayleyth 19388*	Cayley's Theorem (existence version): every group 𝐺 is isomorphic to a subgroup of the symmetric group on the underlying set of 𝐺. (For any group 𝐺 there exists an isomorphism 𝑓 between 𝐺 and a subgroup ℎ of the symmetric group on the underlying set of 𝐺.) See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐻 = (SymGrp‘𝑋)    ⇒   ⊢ (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠)
10.2.10.3  Permutations fixing one element
Theorem	symgfix2 19389*	If a permutation does not move a certain element of a set to a second element, there is a third element which is moved to the second element. (Contributed by AV, 2-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ (𝐿 ∈ 𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))
Theorem	symgextf 19390*	The extension of a permutation, fixing the additional element, is a function. (Contributed by AV, 6-Jan-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸:𝑁⟶𝑁)
Theorem	symgextfv 19391*	The function value of the extension of a permutation, fixing the additional element, for elements in the original domain. (Contributed by AV, 6-Jan-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸‘𝑋) = (𝑍‘𝑋)))
Theorem	symgextfve 19392*	The function value of the extension of a permutation, fixing the additional element, for the additional element. (Contributed by AV, 6-Jan-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ (𝐾 ∈ 𝑁 → (𝑋 = 𝐾 → (𝐸‘𝑋) = 𝐾))
Theorem	symgextf1lem 19393*	Lemma for symgextf1 19394. (Contributed by AV, 6-Jan-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾}) → (𝐸‘𝑋) ≠ (𝐸‘𝑌)))
Theorem	symgextf1 19394*	The extension of a permutation, fixing the additional element, is a 1-1 function. (Contributed by AV, 6-Jan-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸:𝑁–1-1→𝑁)
Theorem	symgextfo 19395*	The extension of a permutation, fixing the additional element, is an onto function. (Contributed by AV, 7-Jan-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸:𝑁–onto→𝑁)
Theorem	symgextf1o 19396*	The extension of a permutation, fixing the additional element, is a bijection. (Contributed by AV, 7-Jan-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸:𝑁–1-1-onto→𝑁)
Theorem	symgextsymg 19397*	The extension of a permutation is an element of the extended symmetric group. (Contributed by AV, 9-Mar-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸 ∈ (Base‘(SymGrp‘𝑁)))
Theorem	symgextres 19398*	The restriction of the extension of a permutation, fixing the additional element, to the original domain. (Contributed by AV, 6-Jan-2019.)
⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝐸 ↾ (𝑁 ∖ {𝐾})) = 𝑍)
Theorem	gsumccatsymgsn 19399	Homomorphic property of composites of permutations with a singleton. (Contributed by AV, 20-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐺 Σg (𝑊 ++ ⟨“𝑍”⟩)) = ((𝐺 Σg 𝑊) ∘ 𝑍))
Theorem	gsmsymgrfixlem1 19400*	Lemma 1 for gsmsymgrfix 19401. (Contributed by AV, 20-Jan-2019.)
⊢ 𝑆 = (SymGrp‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (((𝑊 ∈ Word 𝐵 ∧ 𝑃 ∈ 𝐵) ∧ (𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ (∀𝑖 ∈ (0..^(♯‘𝑊))((𝑊‘𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg 𝑊)‘𝐾) = 𝐾)) → (∀𝑖 ∈ (0..^((♯‘𝑊) + 1))(((𝑊 ++ ⟨“𝑃”⟩)‘𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg (𝑊 ++ ⟨“𝑃”⟩))‘𝐾) = 𝐾))