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Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoppgmndb 18601 Bidirectional form of oppgmnd 18600. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
 
Theoremoppgid 18602 Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
𝑂 = (oppg𝑅)    &    0 = (0g𝑅)        0 = (0g𝑂)
 
Theoremoppggrp 18603 The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Grp → 𝑂 ∈ Grp)
 
Theoremoppggrpb 18604 Bidirectional form of oppggrp 18603. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
 
Theoremoppginv 18605 Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)    &   𝐼 = (invg𝑅)       (𝑅 ∈ Grp → 𝐼 = (invg𝑂))
 
Theoreminvoppggim 18606 The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂))
 
Theoremoppggic 18607 Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ Grp → 𝐺𝑔 𝑂)
 
Theoremoppgsubm 18608 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (SubMnd‘𝐺) = (SubMnd‘𝑂)
 
Theoremoppgsubg 18609 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (SubGrp‘𝐺) = (SubGrp‘𝑂)
 
Theoremoppgcntz 18610 A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑂 = (oppg𝐺)    &   𝑍 = (Cntz‘𝐺)       (𝑍𝐴) = ((Cntz‘𝑂)‘𝐴)
 
Theoremoppgcntr 18611 The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑂 = (oppg𝐺)    &   𝑍 = (Cntr‘𝐺)       𝑍 = (Cntr‘𝑂)
 
Theoremgsumwrev 18612 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 = (oppg𝑀)       ((𝑀 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊)))
 
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 18614. However, the set is allowed to be empty, see symgbas0 18635. 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 18617), 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 18617). This means that a symmetric group is a permutation group, and each permutation group is a subgroup of a symmetric group (see pgrpsubgsymgbi 18654 and pgrpsubgsymg 18655). 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 18657, which is a (proper) subgroup of a symmetric group, see idressubgsymg 18656.

As in [Rotman] p. 28 "Let 𝑥𝑋 and 𝑝 ∈ SymGrp(𝑋); we say 𝑝 fixes 𝑥 if (𝑝𝑥) = 𝑥; otherwise 𝑝 moves 𝑥.". The theorems starting with symgfix2 18662 are about fixed/moved elements.

 
Syntaxcsymg 18613 Extend class notation to include the class of symmetric groups.
class SymGrp
 
Definitiondf-symg 18614* 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 {:𝑥1-1-onto𝑥}))
 
Theoremsymgval 18615* 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 = {𝑥𝑥:𝐴1-1-onto𝐴}       𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
 
TheorempermsetexOLD 18616* Obsolete version of f1osetex 8469 as of 8-Aug-2024. (Contributed by AV, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → {𝑓𝑓:𝐴1-1-onto𝐴} ∈ V)
 
Theoremsymgbas 18617* 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 = {𝑥𝑥:𝐴1-1-onto𝐴}
 
TheoremsymgbasexOLD 18618 Obsolete as of 8-Aug-2024. 𝐵 ∈ V follows immediatly from fvex 6687. (Contributed by AV, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 ∈ V)
 
Theoremelsymgbas2 18619 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)       (𝐹𝑉 → (𝐹𝐵𝐹:𝐴1-1-onto𝐴))
 
Theoremelsymgbas 18620 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)       (𝐴𝑉 → (𝐹𝐵𝐹:𝐴1-1-onto𝐴))
 
Theoremsymgbasf1o 18621 Elements in the symmetric group are 1-1 onto functions. (Contributed by SO, 9-Jul-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝐵𝐹:𝐴1-1-onto𝐴)
 
Theoremsymgbasf 18622 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)       (𝐹𝐵𝐹:𝐴𝐴)
 
Theoremsymgbasmap 18623 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)       (𝐹𝐵𝐹 ∈ (𝐴m 𝐴))
 
Theoremsymghash 18624 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)))
 
Theoremsymgbasfi 18625 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)
 
Theoremsymgfv 18626 The function value of a permutation. (Contributed by AV, 1-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴) → (𝐹𝑋) ∈ 𝐴)
 
Theoremsymgfvne 18627 The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = 𝑍 → (𝑌𝑋 → (𝐹𝑌) ≠ 𝑍)))
 
Theoremsymgressbas 18628 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 = (𝑀s 𝐵)
 
Theoremsymgplusg 18629* 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 = (𝐴m 𝐴)    &    + = (+g𝐺)        + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
 
Theoremsymgov 18630 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)    &    + = (+g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑋𝑌))
 
Theoremsymgcl 18631 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)    &    + = (+g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
Theoremidresperm 18632 The identity function restricted to a set is a permutation of this set. (Contributed by AV, 17-Mar-2019.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))
 
Theoremsymgmov1 18633* 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))       (𝑄𝑃 → ∀𝑛𝑁𝑘𝑁 (𝑄𝑛) = 𝑘)
 
Theoremsymgmov2 18634* 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))       (𝑄𝑃 → ∀𝑛𝑁𝑘𝑁 (𝑄𝑘) = 𝑛)
 
Theoremsymgbas0 18635 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‘∅)) = {∅}
 
Theoremsymg1hash 18636 The symmetric group on a singleton has cardinality 1. (Contributed by AV, 9-Dec-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}       (𝐼𝑉 → (♯‘𝐵) = 1)
 
Theoremsymg1bas 18637 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}       (𝐼𝑉𝐵 = {{⟨𝐼, 𝐼⟩}})
 
Theoremsymg2hash 18638 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}       ((𝐼𝑉𝐽𝑊𝐼𝐽) → (♯‘𝐵) = 2)
 
Theoremsymg2bas 18639 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 18637. (Contributed by AV, 9-Dec-2018.) (Proof shortened by AV, 16-Jun-2022.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼, 𝐽}       ((𝐼𝑉𝐽𝑊) → 𝐵 = {{⟨𝐼, 𝐼⟩, ⟨𝐽, 𝐽⟩}, {⟨𝐼, 𝐽⟩, ⟨𝐽, 𝐼⟩}})
 
Theorem0symgefmndeq 18640 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‘∅)
 
Theoremsnsymgefmndeq 18641 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))
 
Theoremsymgpssefmnd 18642 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)       ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘𝑀))
 
Theoremsymgvalstruct 18643* 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.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}    &   𝑀 = (𝐴m 𝐴)    &    + = (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔))    &   𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))       (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
 
Theoremsymgsubmefmnd 18644 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)       (𝐴𝑉𝐵 ∈ (SubMnd‘𝑀))
 
Theoremsymgtset 18645 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)       (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))
 
Theoremsymggrp 18646 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)       (𝐴𝑉𝐺 ∈ Grp)
 
Theoremsymgid 18647 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)       (𝐴𝑉 → ( I ↾ 𝐴) = (0g𝐺))
 
Theoremsymginv 18648 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 = (invg𝐺)       (𝐹𝐵 → (𝑁𝐹) = 𝐹)
 
TheoremsymgsubmefmndALT 18649 The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. Alternate proof based on issubmndb 18086 and not on injsubmefmnd 18178 and sursubmefmnd 18177. (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)       (𝐴𝑉𝐵 ∈ (SubMnd‘𝑀))
 
Theoremgalactghm 18650* 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 = (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥 𝑦)))       ( ∈ (𝐺 GrpAct 𝑌) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
 
Theoremlactghmga 18651* The converse of galactghm 18650. 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)    &    = (𝑥𝑋, 𝑦𝑌 ↦ ((𝐹𝑥)‘𝑦))       (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∈ (𝐺 GrpAct 𝑌))
 
Theoremsymgtopn 18652 The topology of the symmetric group on 𝐴. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐺 = (SymGrp‘𝑋)    &   𝐵 = (Base‘𝐺)       (𝑋𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺))
 
Theoremsymgga 18653* 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 = (𝑓𝐵, 𝑥𝑋 ↦ (𝑓𝑥))       (𝑋𝑉𝐹 ∈ (𝐺 GrpAct 𝑋))
 
Theorempgrpsubgsymgbi 18654 Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉 → (𝑃 ∈ (SubGrp‘𝐺) ↔ (𝑃𝐵 ∧ (𝐺s 𝑃) ∈ Grp)))
 
Theorempgrpsubgsymg 18655* 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)       (𝐴𝑉 → ((𝑃 ∈ Grp ∧ 𝐹𝐵 ∧ (+g𝑃) = (𝑓𝐹, 𝑔𝐹 ↦ (𝑓𝑔))) → 𝐹 ∈ (SubGrp‘𝐺)))
 
Theoremidressubgsymg 18656 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)       (𝐴𝑉 → {( I ↾ 𝐴)} ∈ (SubGrp‘𝐺))
 
Theoremidrespermg 18657 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 = (𝐺s {( I ↾ 𝐴)})       (𝐴𝑉 → (𝐸 ∈ Grp ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))
 
10.2.10.2  Cayley's theorem
 
Theoremcayleylem1 18658* Lemma for cayley 18660. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐻 = (SymGrp‘𝑋)    &   𝑆 = (Base‘𝐻)    &   𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))       (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻))
 
Theoremcayleylem2 18659* Lemma for cayley 18660. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐻 = (SymGrp‘𝑋)    &   𝑆 = (Base‘𝐻)    &   𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))       (𝐺 ∈ Grp → 𝐹:𝑋1-1𝑆)
 
Theoremcayley 18660* 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)    &    + = (+g𝐺)    &   𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑆 = ran 𝐹       (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐻) ∧ 𝐹 ∈ (𝐺 GrpHom (𝐻s 𝑆)) ∧ 𝐹:𝑋1-1-onto𝑆))
 
Theoremcayleyth 18661* 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 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 (𝐻s 𝑠))𝑓:𝑋1-1-onto𝑠)
 
10.2.10.3  Permutations fixing one element
 
Theoremsymgfix2 18662* 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))       (𝐿𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
 
Theoremsymgextf 18663* The extension of a permutation, fixing the additional element, is a function. (Contributed by AV, 6-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁𝑁)
 
Theoremsymgextfv 18664* 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 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑋) = (𝑍𝑋)))
 
Theoremsymgextfve 18665* 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 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       (𝐾𝑁 → (𝑋 = 𝐾 → (𝐸𝑋) = 𝐾))
 
Theoremsymgextf1lem 18666* Lemma for symgextf1 18667. (Contributed by AV, 6-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → ((𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾}) → (𝐸𝑋) ≠ (𝐸𝑌)))
 
Theoremsymgextf1 18667* 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 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁1-1𝑁)
 
Theoremsymgextfo 18668* 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 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁onto𝑁)
 
Theoremsymgextf1o 18669* The extension of a permutation, fixing the additional element, is a bijection. (Contributed by AV, 7-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁1-1-onto𝑁)
 
Theoremsymgextsymg 18670* 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 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝑁𝑉𝐾𝑁𝑍𝑆) → 𝐸 ∈ (Base‘(SymGrp‘𝑁)))
 
Theoremsymgextres 18671* 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 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → (𝐸 ↾ (𝑁 ∖ {𝐾})) = 𝑍)
 
Theoremgsumccatsymgsn 18672 Homomorphic property of composites of permutations with a singleton. (Contributed by AV, 20-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐴𝑉𝑊 ∈ Word 𝐵𝑍𝐵) → (𝐺 Σg (𝑊 ++ ⟨“𝑍”⟩)) = ((𝐺 Σg 𝑊) ∘ 𝑍))
 
Theoremgsmsymgrfixlem1 18673* Lemma 1 for gsmsymgrfix 18674. (Contributed by AV, 20-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)       (((𝑊 ∈ Word 𝐵𝑃𝐵) ∧ (𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ (∀𝑖 ∈ (0..^(♯‘𝑊))((𝑊𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg 𝑊)‘𝐾) = 𝐾)) → (∀𝑖 ∈ (0..^((♯‘𝑊) + 1))(((𝑊 ++ ⟨“𝑃”⟩)‘𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg (𝑊 ++ ⟨“𝑃”⟩))‘𝐾) = 𝐾))
 
Theoremgsmsymgrfix 18674* The composition of permutations fixing one element also fixes this element. (Contributed by AV, 20-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)       ((𝑁 ∈ Fin ∧ 𝐾𝑁𝑊 ∈ Word 𝐵) → (∀𝑖 ∈ (0..^(♯‘𝑊))((𝑊𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg 𝑊)‘𝐾) = 𝐾))
 
Theoremfvcosymgeq 18675* The values of two compositions of permutations are equal if the values of the composed permutations are pairwise equal. (Contributed by AV, 26-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (SymGrp‘𝑀)    &   𝑃 = (Base‘𝑍)    &   𝐼 = (𝑁𝑀)       ((𝐺𝐵𝐾𝑃) → ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
 
Theoremgsmsymgreqlem1 18676* Lemma 1 for gsmsymgreq 18678. (Contributed by AV, 26-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (SymGrp‘𝑀)    &   𝑃 = (Base‘𝑍)    &   𝐼 = (𝑁𝑀)       (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝐽𝐼) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ∧ (𝐶𝐽) = (𝑅𝐽)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝐽) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝐽)))
 
Theoremgsmsymgreqlem2 18677* Lemma 2 for gsmsymgreq 18678. (Contributed by AV, 26-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (SymGrp‘𝑀)    &   𝑃 = (Base‘𝑍)    &   𝐼 = (𝑁𝑀)       (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
 
Theoremgsmsymgreq 18678* Two combination of permutations moves an element of the intersection of the base sets of the permutations to the same element if each pair of corresponding permutations moves such an element to the same element. (Contributed by AV, 20-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (SymGrp‘𝑀)    &   𝑃 = (Base‘𝑍)    &   𝐼 = (𝑁𝑀)       (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝑊 ∈ Word 𝐵𝑈 ∈ Word 𝑃 ∧ (♯‘𝑊) = (♯‘𝑈))) → (∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛𝐼 ((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)))
 
Theoremsymgfixelq 18679* A permutation of a set fixing an element of the set. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}       (𝐹𝑉 → (𝐹𝑄 ↔ (𝐹:𝑁1-1-onto𝑁 ∧ (𝐹𝐾) = 𝐾)))
 
Theoremsymgfixels 18680* The restriction of a permutation to a set with one element removed is an element of the restricted symmetric group if the restriction is a 1-1 onto function. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐷 = (𝑁 ∖ {𝐾})       (𝐹𝑉 → ((𝐹𝐷) ∈ 𝑆 ↔ (𝐹𝐷):𝐷1-1-onto𝐷))
 
Theoremsymgfixelsi 18681* The restriction of a permutation fixing an element to the set with this element removed is an element of the restricted symmetric group. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐷 = (𝑁 ∖ {𝐾})       ((𝐾𝑁𝐹𝑄) → (𝐹𝐷) ∈ 𝑆)
 
Theoremsymgfixf 18682* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a function. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))       (𝐾𝑁𝐻:𝑄𝑆)
 
Theoremsymgfixf1 18683* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a 1-1 function. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))       (𝐾𝑁𝐻:𝑄1-1𝑆)
 
Theoremsymgfixfolem1 18684* Lemma 1 for symgfixfo 18685. (Contributed by AV, 7-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝑁𝑉𝐾𝑁𝑍𝑆) → 𝐸𝑄)
 
Theoremsymgfixfo 18685* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is an onto function. (Contributed by AV, 7-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))       ((𝑁𝑉𝐾𝑁) → 𝐻:𝑄onto𝑆)
 
Theoremsymgfixf1o 18686* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a bijection. (Contributed by AV, 7-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))       ((𝑁𝑉𝐾𝑁) → 𝐻:𝑄1-1-onto𝑆)
 
10.2.10.4  Transpositions in the symmetric group

Transpositions are special cases of "cycles" as defined in [Rotman] p. 28: "Let i1 , i2 , ... , ir be distinct integers between 1 and n. If α in Sn fixes the other integers and α(i1) = i2, α(i2) = i3, ..., α(ir-1 ) = ir, α(ir) = i1, then α is an r-cycle. We also say that α is a cycle of length r." and in [Rotman] p. 31: "A 2-cycle is also called transposition.".

We (currently) do not have/need a definition for cycles, so transpositions are explicitly defined in df-pmtr 18688.

 
Syntaxcpmtr 18687 Syntax for the transposition generator function.
class pmTrsp
 
Definitiondf-pmtr 18688* Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
 
Theoremf1omvdmvd 18689 A permutation of any class moves a point which is moved to a different point which is moved. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))
 
Theoremf1omvdcnv 18690 A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
(𝐹:𝐴1-1-onto𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ))
 
Theoremmvdco 18691 Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
dom ((𝐹𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
 
Theoremf1omvdconj 18692 Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝐹:𝐴𝐴𝐺:𝐴1-1-onto𝐴) → dom (((𝐺𝐹) ∘ 𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))
 
Theoremf1otrspeq 18693 A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
(((𝐹:𝐴1-1-onto𝐴𝐺:𝐴1-1-onto𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → 𝐹 = 𝐺)
 
Theoremf1omvdco2 18694 If exactly one of two permutations is limited to a set of points, then the composition will not be. (Contributed by Stefan O'Rear, 23-Aug-2015.)
((𝐹:𝐴1-1-onto𝐴𝐺:𝐴1-1-onto𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹𝐺) ∖ I ) ⊆ 𝑋)
 
Theoremf1omvdco3 18695 If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)
((𝐹:𝐴1-1-onto𝐴𝐺:𝐴1-1-onto𝐴 ∧ (𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I ))) → 𝑋 ∈ dom ((𝐹𝐺) ∖ I ))
 
Theorempmtrfval 18696* The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
 
Theorempmtrval 18697* A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
 
Theorempmtrfv 18698 General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
 
Theorempmtrprfv 18699 In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑋) = 𝑌)
 
Theorempmtrprfv3 18700 In a transposition of two given points, all other points are mapped to themselves. (Contributed by AV, 17-Mar-2019.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → ((𝑇‘{𝑋, 𝑌})‘𝑍) = 𝑍)
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