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	symgfixelq 19301*	A permutation of a set fixing an element of the set. (Contributed by AV, 4-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑄 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝑄 ↔ (𝐹:𝑁–1-1-onto→𝑁 ∧ (𝐹‘𝐾) = 𝐾)))
Theorem	symgfixels 19302*	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 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}    &   ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐷 = (𝑁 ∖ {𝐾})    ⇒   ⊢ (𝐹 ∈ 𝑉 → ((𝐹 ↾ 𝐷) ∈ 𝑆 ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→𝐷))
Theorem	symgfixelsi 19303*	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 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}    &   ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐷 = (𝑁 ∖ {𝐾})    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐹 ∈ 𝑄) → (𝐹 ↾ 𝐷) ∈ 𝑆)
Theorem	symgfixf 19304*	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 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}    &   ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐻 = (𝑞 ∈ 𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))    ⇒   ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄⟶𝑆)
Theorem	symgfixf1 19305*	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 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}    &   ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐻 = (𝑞 ∈ 𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))    ⇒   ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄–1-1→𝑆)
Theorem	symgfixfolem1 19306*	Lemma 1 for symgfixfo 19307. (Contributed by AV, 7-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑄 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}    &   ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐻 = (𝑞 ∈ 𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))    &   ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))    ⇒   ⊢ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸 ∈ 𝑄)
Theorem	symgfixfo 19307*	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 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}    &   ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐻 = (𝑞 ∈ 𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))    ⇒   ⊢ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝐻:𝑄–onto→𝑆)
Theorem	symgfixf1o 19308*	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 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}    &   ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝐻 = (𝑞 ∈ 𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))    ⇒   ⊢ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝐻:𝑄–1-1-onto→𝑆)
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 19310.
Syntax	cpmtr 19309	Syntax for the transposition generator function.
class pmTrsp
Definition	df-pmtr 19310*	Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
Theorem	f1omvdmvd 19311	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:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))
Theorem	f1omvdcnv 19312	A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))
Theorem	mvdco 19313	Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
Theorem	f1omvdconj 19314	Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))
Theorem	f1otrspeq 19315	A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → 𝐹 = 𝐺)
Theorem	f1omvdco2 19316	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:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)
Theorem	f1omvdco3 19317	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:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴 ∧ (𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I ))) → 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))
Theorem	pmtrfval 19318*	The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    ⇒   ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
Theorem	pmtrval 19319*	A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
Theorem	pmtrfv 19320	General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    ⇒   ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))
Theorem	pmtrprfv 19321	In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑋) = 𝑌)
Theorem	pmtrprfv3 19322	In a transposition of two given points, all other points are mapped to themselves. (Contributed by AV, 17-Mar-2019.)
⊢ 𝑇 = (pmTrsp‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑍 ∈ 𝐷) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → ((𝑇‘{𝑋, 𝑌})‘𝑍) = 𝑍)
Theorem	pmtrf 19323	Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷)
Theorem	pmtrmvd 19324	A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → dom ((𝑇‘𝑃) ∖ I ) = 𝑃)
Theorem	pmtrrn 19325	Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ 𝑅)
Theorem	pmtrfrn 19326	A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    &   ⊢ 𝑃 = dom (𝐹 ∖ I )    ⇒   ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃)))
Theorem	pmtrffv 19327	Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    &   ⊢ 𝑃 = dom (𝐹 ∖ I )    ⇒   ⊢ ((𝐹 ∈ 𝑅 ∧ 𝑍 ∈ 𝐷) → (𝐹‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))
Theorem	pmtrrn2 19328*	For any transposition there are two points it is transposing. (Contributed by SO, 15-Jul-2018.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    ⇒   ⊢ (𝐹 ∈ 𝑅 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 (𝑥 ≠ 𝑦 ∧ 𝐹 = (𝑇‘{𝑥, 𝑦})))
Theorem	pmtrfinv 19329	A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    ⇒   ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷))
Theorem	pmtrfmvdn0 19330	A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    ⇒   ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅)
Theorem	pmtrff1o 19331	A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    ⇒   ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷–1-1-onto→𝐷)
Theorem	pmtrfcnv 19332	A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    ⇒   ⊢ (𝐹 ∈ 𝑅 → ◡𝐹 = 𝐹)
Theorem	pmtrfb 19333	An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    ⇒   ⊢ (𝐹 ∈ 𝑅 ↔ (𝐷 ∈ V ∧ 𝐹:𝐷–1-1-onto→𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o))
Theorem	pmtrfconj 19334	Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝑅 = ran 𝑇    ⇒   ⊢ ((𝐹 ∈ 𝑅 ∧ 𝐺:𝐷–1-1-onto→𝐷) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∈ 𝑅)
Theorem	symgsssg 19335*	The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ⊆ 𝑋} ∈ (SubGrp‘𝐺))
Theorem	symgfisg 19336*	The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺))
Theorem	symgtrf 19337	Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ 𝑇 ⊆ 𝐵
Theorem	symggen 19338*	The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺))    ⇒   ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
Theorem	symggen2 19339	A finite permutation group is generated by the transpositions, see also Theorem 3.4 in [Rotman] p. 31. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺))    ⇒   ⊢ (𝐷 ∈ Fin → (𝐾‘𝑇) = 𝐵)
Theorem	symgtrinv 19340	To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼‘(𝐺 Σg 𝑊)) = (𝐺 Σg (reverse‘𝑊)))
Theorem	pmtr3ncomlem1 19341	Lemma 1 for pmtr3ncom 19343. (Contributed by AV, 17-Mar-2018.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝐹 = (𝑇‘{𝑋, 𝑌})    &   ⊢ 𝐺 = (𝑇‘{𝑌, 𝑍})    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑍 ∈ 𝐷) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → ((𝐺 ∘ 𝐹)‘𝑋) ≠ ((𝐹 ∘ 𝐺)‘𝑋))
Theorem	pmtr3ncomlem2 19342	Lemma 2 for pmtr3ncom 19343. (Contributed by AV, 17-Mar-2018.)
⊢ 𝑇 = (pmTrsp‘𝐷)    &   ⊢ 𝐹 = (𝑇‘{𝑋, 𝑌})    &   ⊢ 𝐺 = (𝑇‘{𝑌, 𝑍})    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑍 ∈ 𝐷) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (𝐺 ∘ 𝐹) ≠ (𝐹 ∘ 𝐺))
Theorem	pmtr3ncom 19343*	Transpositions over sets with at least 3 elements are not commutative, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
⊢ 𝑇 = (pmTrsp‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐷)) → ∃𝑓 ∈ ran 𝑇∃𝑔 ∈ ran 𝑇(𝑔 ∘ 𝑓) ≠ (𝑓 ∘ 𝑔))
Theorem	pmtrdifellem1 19344	Lemma 1 for pmtrdifel 19348. (Contributed by AV, 15-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    &   ⊢ 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))    ⇒   ⊢ (𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅)
Theorem	pmtrdifellem2 19345	Lemma 2 for pmtrdifel 19348. (Contributed by AV, 15-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    &   ⊢ 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))    ⇒   ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
Theorem	pmtrdifellem3 19346*	Lemma 3 for pmtrdifel 19348. (Contributed by AV, 15-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    &   ⊢ 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))    ⇒   ⊢ (𝑄 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑥) = (𝑆‘𝑥))
Theorem	pmtrdifellem4 19347	Lemma 4 for pmtrdifel 19348. (Contributed by AV, 28-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    &   ⊢ 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))    ⇒   ⊢ ((𝑄 ∈ 𝑇 ∧ 𝐾 ∈ 𝑁) → (𝑆‘𝐾) = 𝐾)
Theorem	pmtrdifel 19348*	A transposition of elements of a set without a special element corresponds to a transposition of elements of the set. (Contributed by AV, 15-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    ⇒   ⊢ ∀𝑡 ∈ 𝑇 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥)
Theorem	pmtrdifwrdellem1 19349*	Lemma 1 for pmtrdifwrdel 19353. (Contributed by AV, 15-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    &   ⊢ 𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑥) ∖ I )))    ⇒   ⊢ (𝑊 ∈ Word 𝑇 → 𝑈 ∈ Word 𝑅)
Theorem	pmtrdifwrdellem2 19350*	Lemma 2 for pmtrdifwrdel 19353. (Contributed by AV, 15-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    &   ⊢ 𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑥) ∖ I )))    ⇒   ⊢ (𝑊 ∈ Word 𝑇 → (♯‘𝑊) = (♯‘𝑈))
Theorem	pmtrdifwrdellem3 19351*	Lemma 3 for pmtrdifwrdel 19353. (Contributed by AV, 15-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    &   ⊢ 𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑥) ∖ I )))    ⇒   ⊢ (𝑊 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))
Theorem	pmtrdifwrdel2lem1 19352*	Lemma 1 for pmtrdifwrdel2 19354. (Contributed by AV, 31-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    &   ⊢ 𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑥) ∖ I )))    ⇒   ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐾 ∈ 𝑁) → ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈‘𝑖)‘𝐾) = 𝐾)
Theorem	pmtrdifwrdel 19353*	A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set. (Contributed by AV, 15-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    ⇒   ⊢ ∀𝑤 ∈ Word 𝑇∃𝑢 ∈ Word 𝑅((♯‘𝑤) = (♯‘𝑢) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))
Theorem	pmtrdifwrdel2 19354*	A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set not moving the special element. (Contributed by AV, 31-Jan-2019.)
⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   ⊢ 𝑅 = ran (pmTrsp‘𝑁)    ⇒   ⊢ (𝐾 ∈ 𝑁 → ∀𝑤 ∈ Word 𝑇∃𝑢 ∈ Word 𝑅((♯‘𝑤) = (♯‘𝑢) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))(((𝑢‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))))
Theorem	pmtrprfval 19355*	The transpositions on a pair. (Contributed by AV, 9-Dec-2018.)
⊢ (pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
Theorem	pmtrprfvalrn 19356	The range of the transpositions on a pair is actually a singleton: the transposition of the two elements of the pair. (Contributed by AV, 9-Dec-2018.)
⊢ ran (pmTrsp‘{1, 2}) = {{⟨1, 2⟩, ⟨2, 1⟩}}
10.2.10.5  The sign of a permutation
Syntax	cpsgn 19357	Syntax for the sign of a permutation.
class pmSgn
Syntax	cevpm 19358	Syntax for even permutations.
class pmEven
Definition	df-psgn 19359*	Define a function which takes the value 1 for even permutations and -1 for odd. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
Definition	df-evpm 19360	Define the set of even permutations on a given set. (Contributed by Stefan O'Rear, 9-Jul-2018.)
⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1}))
Theorem	psgnunilem1 19361*	Lemma for psgnuni 19367. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑇)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ dom (𝑃 ∖ I ))    ⇒   ⊢ (𝜑 → ((𝑃 ∘ 𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟 ∈ 𝑇 ∃𝑠 ∈ 𝑇 ((𝑃 ∘ 𝑄) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
Theorem	psgnunilem5 19362*	Lemma for psgnuni 19367. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving 𝐴 in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Proof shortened by AV, 12-Oct-2022.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)    &   ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   ⊢ (𝜑 → (♯‘𝑊) = 𝐿)    &   ⊢ (𝜑 → 𝐼 ∈ (0..^𝐿))    &   ⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ))    ⇒   ⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))
Theorem	psgnunilem2 19363*	Lemma for psgnuni 19367. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)    &   ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   ⊢ (𝜑 → (♯‘𝑊) = 𝐿)    &   ⊢ (𝜑 → 𝐼 ∈ (0..^𝐿))    &   ⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ))    &   ⊢ (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))    ⇒   ⊢ (𝜑 → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ))))
Theorem	psgnunilem3 19364*	Lemma for psgnuni 19367. Any nonempty representation of the identity can be incrementally transformed into a representation two shorter. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝐿)    &   ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)    &   ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   ⊢ (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))    ⇒   ⊢ ¬ 𝜑
Theorem	psgnunilem4 19365	Lemma for psgnuni 19367. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)    &   ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    ⇒   ⊢ (𝜑 → (-1↑(♯‘𝑊)) = 1)
Theorem	m1expaddsub 19366	Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = (-1↑(𝑋 + 𝑌)))
Theorem	psgnuni 19367	If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ Word 𝑇)    &   ⊢ (𝜑 → (𝐺 Σg 𝑊) = (𝐺 Σg 𝑋))    ⇒   ⊢ (𝜑 → (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑋)))
Theorem	psgnfval 19368*	Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ 𝑁 = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
Theorem	psgnfn 19369*	Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ 𝑁 Fn 𝐹
Theorem	psgndmsubg 19370	The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 ∈ (SubGrp‘𝐺))
Theorem	psgneldm 19371	Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑃 ∈ dom 𝑁 ↔ (𝑃 ∈ 𝐵 ∧ dom (𝑃 ∖ I ) ∈ Fin))
Theorem	psgneldm2 19372*	The finitary permutations are the span of the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σg 𝑤)))
Theorem	psgneldm2i 19373	A sequence of transpositions describes a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐺 Σg 𝑊) ∈ dom 𝑁)
Theorem	psgneu 19374*	A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑃 ∈ dom 𝑁 → ∃!𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))
Theorem	psgnval 19375*	Value of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑃 ∈ dom 𝑁 → (𝑁‘𝑃) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
Theorem	psgnvali 19376*	A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑃 ∈ dom 𝑁 → ∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ (𝑁‘𝑃) = (-1↑(♯‘𝑤))))
Theorem	psgnvalii 19377	Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝑁‘(𝐺 Σg 𝑊)) = (-1↑(♯‘𝑊)))
Theorem	psgnpmtr 19378	All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑃 ∈ 𝑇 → (𝑁‘𝑃) = -1)
Theorem	psgn0fv0 19379	The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
⊢ ((pmSgn‘∅)‘∅) = 1
Theorem	sygbasnfpfi 19380	The class of non-fixed points of a permutation of a finite set is finite. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → dom (𝑃 ∖ I ) ∈ Fin)
Theorem	psgnfvalfi 19381*	Function definition of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝐷 ∈ Fin → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
Theorem	psgnvalfi 19382*	Value of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → (𝑁‘𝑃) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
Theorem	psgnran 19383	The range of the permutation sign function for finite permutations. (Contributed by AV, 1-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1})
Theorem	gsmtrcl 19384	The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 19373. (Contributed by AV, 19-Jan-2019.)
⊢ 𝑆 = (SymGrp‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑇 = ran (pmTrsp‘𝑁)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑊 ∈ Word 𝑇) → (𝑆 Σg 𝑊) ∈ 𝐵)
Theorem	psgnfitr 19385*	A permutation of a finite set is generated by transpositions. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝑁)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝑁)    ⇒   ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝐵 ↔ ∃𝑤 ∈ Word 𝑇𝑄 = (𝐺 Σg 𝑤)))
Theorem	psgnfieu 19386*	A permutation of a finite set has exactly one parity. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝑁)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝑁)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝐵) → ∃!𝑠∃𝑤 ∈ Word 𝑇(𝑄 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))
Theorem	pmtrsn 19387	The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)
⊢ (pmTrsp‘{𝐴}) = ∅
Theorem	psgnsn 19388	The permutation sign function for a singleton. (Contributed by AV, 6-Aug-2019.)
⊢ 𝐷 = {𝐴}    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1)
Theorem	psgnprfval 19389*	The permutation sign function for a pair. (Contributed by AV, 10-Dec-2018.)
⊢ 𝐷 = {1, 2}    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑋 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
Theorem	psgnprfval1 19390	The permutation sign of the identity for a pair. (Contributed by AV, 11-Dec-2018.)
⊢ 𝐷 = {1, 2}    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑁‘{⟨1, 1⟩, ⟨2, 2⟩}) = 1
Theorem	psgnprfval2 19391	The permutation sign of the transposition for a pair. (Contributed by AV, 10-Dec-2018.)
⊢ 𝐷 = {1, 2}    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑁‘{⟨1, 2⟩, ⟨2, 1⟩}) = -1
10.2.11  p-Groups and Sylow groups; Sylow's theorems
Syntax	cod 19392	Extend class notation to include the order function on the elements of a group.
class od
Syntax	cgex 19393	Extend class notation to include the order function on the elements of a group.
class gEx
Syntax	cpgp 19394	Extend class notation to include the class of all p-groups.
class pGrp
Syntax	cslw 19395	Extend class notation to include the class of all Sylow p-subgroups of a group.
class pSyl
Definition	df-od 19396*	Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) (Revised by AV, 5-Oct-2020.)
⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
Definition	df-gex 19397*	Define the exponent of a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) (Revised by AV, 26-Sep-2020.)
⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
Definition	df-pgp 19398*	Define the set of p-groups, which are groups such that every element has a power of 𝑝 as its order. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by AV, 5-Oct-2020.)
⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
Definition	df-slw 19399*	Define the set of Sylow p-subgroups of a group 𝑔. A Sylow p-subgroup is a p-group that is not a subgroup of any other p-groups in 𝑔. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)})
Theorem	odfval 19400*	Value of the order function. For a shorter proof using ax-rep 5286, see odfvalALT 19401. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove dependency on ax-rep 5286. (Revised by Rohan Ridenour, 17-Aug-2023.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))