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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsymg2bas 18201 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 18199. (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⟩}})

Theoremsymgtset 18202 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.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))

Theoremsymggrp 18203 The symmetric group on a set 𝐴 is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉𝐺 ∈ Grp)

Theoremsymgid 18204 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.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉 → ( I ↾ 𝐴) = (0g𝐺))

Theoremsymginv 18205 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)

Theoremgalactghm 18206* 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 18207* The converse of galactghm 18206. 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 18208 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 18209* 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 18210 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 18211* Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐹 = (Base‘𝑃)       (𝐴𝑉 → ((𝑃 ∈ Grp ∧ 𝐹𝐵 ∧ (+g𝑃) = (𝑓𝐹, 𝑔𝐹 ↦ (𝑓𝑔))) → 𝐹 ∈ (SubGrp‘𝐺)))

Theoremidresperm 18212 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))

Theoremidressubgsymg 18213 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 18214 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)))

10.2.9.2  Cayley's theorem

Theoremcayleylem1 18215* Lemma for cayley 18217. (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 18216* Lemma for cayley 18217. (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 18217* 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 18218* 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)

10.2.9.3  Permutations fixing one element

Theoremsymgfix2 18219* 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 18220* 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 18221* 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 18222* 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 18223* Lemma for symgextf1 18224. (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 18224* 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 18225* 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 18226* 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 18227* 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 18228* 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 18229 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 18230* Lemma 1 for gsmsymgrfix 18231. (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 18231* 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 18232* 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 18233* Lemma 1 for gsmsymgreq 18235. (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 18234* Lemma 2 for gsmsymgreq 18235. (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 18235* 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 18236* 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 18237* 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 18238* 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 18239* 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 18240* 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 18241* Lemma 1 for symgfixfo 18242. (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 18242* 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 18243* 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)

10.2.9.4  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 18245.

Syntaxcpmtr 18244 Syntax for the transposition generator function.
class pmTrsp

Definitiondf-pmtr 18245* 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 18246 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 18247 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 18248 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 18249 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 18250 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 18251 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 18252 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 18253* 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 18254* 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 18255 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 18256 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 18257 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)

Theorempmtrf 18258 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃):𝐷𝐷)

Theorempmtrmvd 18259 A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → dom ((𝑇𝑃) ∖ I ) = 𝑃)

Theorempmtrrn 18260 Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) ∈ 𝑅)

Theorempmtrfrn 18261 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 )       (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃)))

Theorempmtrffv 18262 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 )       ((𝐹𝑅𝑍𝐷) → (𝐹𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))

Theorempmtrrn2 18263* For any transposition there are two points it is transposing. (Contributed by SO, 15-Jul-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝐹 = (𝑇‘{𝑥, 𝑦})))

Theorempmtrfinv 18264 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → (𝐹𝐹) = ( I ↾ 𝐷))

Theorempmtrfmvdn0 18265 A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → dom (𝐹 ∖ I ) ≠ ∅)

Theorempmtrff1o 18266 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅𝐹:𝐷1-1-onto𝐷)

Theorempmtrfcnv 18267 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅𝐹 = 𝐹)

Theorempmtrfb 18268 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 ↔ (𝐷 ∈ V ∧ 𝐹:𝐷1-1-onto𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o))

Theorempmtrfconj 18269 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       ((𝐹𝑅𝐺:𝐷1-1-onto𝐷) → ((𝐺𝐹) ∘ 𝐺) ∈ 𝑅)

Theoremsymgsssg 18270* 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)       (𝐷𝑉 → {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ⊆ 𝑋} ∈ (SubGrp‘𝐺))

Theoremsymgfisg 18271* 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)       (𝐷𝑉 → {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺))

Theoremsymgtrf 18272 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)       𝑇𝐵

Theoremsymggen 18273* 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))       (𝐷𝑉 → (𝐾𝑇) = {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})

Theoremsymggen2 18274 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 → (𝐾𝑇) = 𝐵)

Theoremsymgtrinv 18275 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 = (invg𝐺)       ((𝐷𝑉𝑊 ∈ Word 𝑇) → (𝐼‘(𝐺 Σg 𝑊)) = (𝐺 Σg (reverse‘𝑊)))

Theorempmtr3ncomlem1 18276 Lemma 1 for pmtr3ncom 18278. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → ((𝐺𝐹)‘𝑋) ≠ ((𝐹𝐺)‘𝑋))

Theorempmtr3ncomlem2 18277 Lemma 2 for pmtr3ncom 18278. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (𝐺𝐹) ≠ (𝐹𝐺))

Theorempmtr3ncom 18278* 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)       ((𝐷𝑉 ∧ 3 ≤ (♯‘𝐷)) → ∃𝑓 ∈ ran 𝑇𝑔 ∈ ran 𝑇(𝑔𝑓) ≠ (𝑓𝑔))

Theorempmtrdifellem1 18279 Lemma 1 for pmtrdifel 18283. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇𝑆𝑅)

Theorempmtrdifellem2 18280 Lemma 2 for pmtrdifel 18283. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))

Theorempmtrdifellem3 18281* Lemma 3 for pmtrdifel 18283. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄𝑥) = (𝑆𝑥))

Theorempmtrdifellem4 18282 Lemma 4 for pmtrdifel 18283. (Contributed by AV, 28-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       ((𝑄𝑇𝐾𝑁) → (𝑆𝐾) = 𝐾)

Theorempmtrdifel 18283* 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)       𝑡𝑇𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥)

Theorempmtrdifwrdellem1 18284* Lemma 1 for pmtrdifwrdel 18288. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇𝑈 ∈ Word 𝑅)

Theorempmtrdifwrdellem2 18285* Lemma 2 for pmtrdifwrdel 18288. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇 → (♯‘𝑊) = (♯‘𝑈))

Theorempmtrdifwrdellem3 18286* Lemma 3 for pmtrdifwrdel 18288. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))

Theorempmtrdifwrdel2lem1 18287* Lemma 1 for pmtrdifwrdel2 18289. (Contributed by AV, 31-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       ((𝑊 ∈ Word 𝑇𝐾𝑁) → ∀𝑖 ∈ (0..^(♯‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾)

Theorempmtrdifwrdel 18288* 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 𝑇𝑢 ∈ Word 𝑅((♯‘𝑤) = (♯‘𝑢) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥))

Theorempmtrdifwrdel2 18289* 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)       (𝐾𝑁 → ∀𝑤 ∈ Word 𝑇𝑢 ∈ Word 𝑅((♯‘𝑤) = (♯‘𝑢) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))(((𝑢𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥))))

Theorempmtrprfval 18290* 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)))

Theorempmtrprfvalrn 18291 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⟩}}

10.2.9.5  The sign of a permutation

Syntaxcpsgn 18292 Syntax for the sign of a permutation.
class pmSgn

Syntaxcevpm 18293 Syntax for even permutations.
class pmEven

Definitiondf-psgn 18294* 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} ↦ (℩𝑠𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))

Definitiondf-evpm 18295 Define the set of even permutations on a given set. (Contributed by Stefan O'Rear, 9-Jul-2018.)
pmEven = (𝑑 ∈ V ↦ ((pmSgn‘𝑑) “ {1}))

Theorempsgnunilem1 18296* Lemma for psgnuni 18303. 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)    &   (𝜑𝐷𝑉)    &   (𝜑𝑃𝑇)    &   (𝜑𝑄𝑇)    &   (𝜑𝐴 ∈ dom (𝑃 ∖ I ))       (𝜑 → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))

Theorempsgnunilem5 18297* Lemma for psgnuni 18303. 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)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → (♯‘𝑊) = 𝐿)    &   (𝜑𝐼 ∈ (0..^𝐿))    &   (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))    &   (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ))       (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))

Theorempsgnunilem5OLD 18298* Obsolete version of psgnunilem5 18297 as of 12-Oct-2022. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → (♯‘𝑊) = 𝐿)    &   (𝜑𝐼 ∈ (0..^𝐿))    &   (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))    &   (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ))       (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))

Theorempsgnunilem2 18299* Lemma for psgnuni 18303. 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)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → (♯‘𝑊) = 𝐿)    &   (𝜑𝐼 ∈ (0..^𝐿))    &   (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))    &   (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ 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 ((𝑤𝑗) ∖ I ))))

Theorempsgnunilem3 18300* Lemma for psgnuni 18303. 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)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ 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

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