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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elsymgbas 18501 | 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 18502 | 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 18503 | 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 18504 | 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 18505 | 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 18506 | 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 18507 | 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 18508 | 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 18509 | 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 18510* | 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.) |
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (𝐴 ↑m 𝐴) & ⊢ + = (+g‘𝐺) ⇒ ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) | ||
Theorem | symgov 18511 | 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 18512 | 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 18513 | 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 18514* | 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 18515* | 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 18516 | 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 18517 | 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 18518 | 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 18519 | 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 18520 | 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 18518. (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 18521 | 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 18522 | 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 18523 | 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 18524* | 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 = {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 18525 | 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 18526 | 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 18527 | 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 18528 | 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 18529 | 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 18530 | The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. Alternate proof based on issubmndb 17969 and not on injsubmefmnd 18061 and sursubmefmnd 18060. (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 18531* | 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 18532* | The converse of galactghm 18531. 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 18533 | 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 18534* | 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 18535 | 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 18536* | 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 18537 | 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 18538 | 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))) | ||
Theorem | cayleylem1 18539* | Lemma for cayley 18541. (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 18540* | Lemma for cayley 18541. (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 18541* | 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 18542* | 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) | ||
Theorem | symgfix2 18543* | 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 18544* | 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 18545* | 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 18546* | 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 18547* | Lemma for symgextf1 18548. (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 18548* | 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 18549* | 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 18550* | 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 18551* | 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 18552* | 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 18553 | 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 18554* | Lemma 1 for gsmsymgrfix 18555. (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)) | ||
Theorem | gsmsymgrfix 18555* | 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 ∧ 𝐾 ∈ 𝑁 ∧ 𝑊 ∈ Word 𝐵) → (∀𝑖 ∈ (0..^(♯‘𝑊))((𝑊‘𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg 𝑊)‘𝐾) = 𝐾)) | ||
Theorem | fvcosymgeq 18556* | 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 = (𝑁 ∩ 𝑀) ⇒ ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐾 ∈ 𝑃) → ((𝑋 ∈ 𝐼 ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑛 ∈ 𝐼 (𝐹‘𝑛) = (𝐻‘𝑛)) → ((𝐹 ∘ 𝐺)‘𝑋) = ((𝐻 ∘ 𝐾)‘𝑋))) | ||
Theorem | gsmsymgreqlem1 18557* | Lemma 1 for gsmsymgreq 18559. (Contributed by AV, 26-Jan-2019.) |
⊢ 𝑆 = (SymGrp‘𝑁) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑍 = (SymGrp‘𝑀) & ⊢ 𝑃 = (Base‘𝑍) & ⊢ 𝐼 = (𝑁 ∩ 𝑀) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝐽 ∈ 𝐼) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ∧ (𝐶‘𝐽) = (𝑅‘𝐽)) → ((𝑆 Σg (𝑋 ++ 〈“𝐶”〉))‘𝐽) = ((𝑍 Σg (𝑌 ++ 〈“𝑅”〉))‘𝐽))) | ||
Theorem | gsmsymgreqlem2 18558* | Lemma 2 for gsmsymgreq 18559. (Contributed by AV, 26-Jan-2019.) |
⊢ 𝑆 = (SymGrp‘𝑁) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑍 = (SymGrp‘𝑀) & ⊢ 𝑃 = (Base‘𝑍) & ⊢ 𝐼 = (𝑁 ∩ 𝑀) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ 〈“𝐶”〉)))∀𝑛 ∈ 𝐼 (((𝑋 ++ 〈“𝐶”〉)‘𝑖)‘𝑛) = (((𝑌 ++ 〈“𝑅”〉)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ 〈“𝐶”〉))‘𝑛) = ((𝑍 Σg (𝑌 ++ 〈“𝑅”〉))‘𝑛)))) | ||
Theorem | gsmsymgreq 18559* | 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 = (𝑁 ∩ 𝑀) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝑊 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝑃 ∧ (♯‘𝑊) = (♯‘𝑈))) → (∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ 𝐼 ((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛))) | ||
Theorem | symgfixelq 18560* | 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 18561* | 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 18562* | 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 18563* | 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 18564* | 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 18565* | Lemma 1 for symgfixfo 18566. (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 18566* | 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 18567* | 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) | ||
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.".
| ||
Syntax | cpmtr 18568 | Syntax for the transposition generator function. |
class pmTrsp | ||
Definition | df-pmtr 18569* | 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 18570 | 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 18571 | 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 18572 | 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 18573 | 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 18574 | 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 18575 | 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 18576 | 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 18577* | 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 18578* | 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 18579 | 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 18580 | 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 18581 | 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 18582 | 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 18583 | 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 18584 | 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 18585 | 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 18586 | 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 18587* | 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 18588 | 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 18589 | 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 18590 | 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 18591 | 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 18592 | 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 18593 | 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 18594* | 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 18595* | 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 18596 | 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 18597* | 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 18598 | 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 18599 | 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 18600 | Lemma 1 for pmtr3ncom 18602. (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)) |
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