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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oppgplusfval 17601 | Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
⊢ + = (+g‘𝑅) & ⊢ 𝑂 = (oppg‘𝑅) & ⊢ ✚ = (+g‘𝑂) ⇒ ⊢ ✚ = tpos + | ||
Theorem | oppgplus 17602 | Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
⊢ + = (+g‘𝑅) & ⊢ 𝑂 = (oppg‘𝑅) & ⊢ ✚ = (+g‘𝑂) ⇒ ⊢ (𝑋 ✚ 𝑌) = (𝑌 + 𝑋) | ||
Theorem | oppglem 17603 | Lemma for oppgbas 17604. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ 𝑂 = (oppg‘𝑅) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 ≠ 2 ⇒ ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) | ||
Theorem | oppgbas 17604 | Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ 𝑂 = (oppg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
Theorem | oppgtset 17605 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
⊢ 𝑂 = (oppg‘𝑅) & ⊢ 𝐽 = (TopSet‘𝑅) ⇒ ⊢ 𝐽 = (TopSet‘𝑂) | ||
Theorem | oppgtopn 17606 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
⊢ 𝑂 = (oppg‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑅) ⇒ ⊢ 𝐽 = (TopOpen‘𝑂) | ||
Theorem | oppgmnd 17607 | The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) |
⊢ 𝑂 = (oppg‘𝑅) ⇒ ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd) | ||
Theorem | oppgmndb 17608 | Bidirectional form of oppgmnd 17607. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ 𝑂 = (oppg‘𝑅) ⇒ ⊢ (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd) | ||
Theorem | oppgid 17609 | Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) |
⊢ 𝑂 = (oppg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ 0 = (0g‘𝑂) | ||
Theorem | oppggrp 17610 | The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ 𝑂 = (oppg‘𝑅) ⇒ ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) | ||
Theorem | oppggrpb 17611 | Bidirectional form of oppggrp 17610. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ 𝑂 = (oppg‘𝑅) ⇒ ⊢ (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp) | ||
Theorem | oppginv 17612 | Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ 𝑂 = (oppg‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) ⇒ ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂)) | ||
Theorem | invoppggim 17613 | The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ 𝑂 = (oppg‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂)) | ||
Theorem | oppggic 17614 | Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ 𝑂 = (oppg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝐺 ≃𝑔 𝑂) | ||
Theorem | oppgsubm 17615 | Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
⊢ 𝑂 = (oppg‘𝐺) ⇒ ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂) | ||
Theorem | oppgsubg 17616 | Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
⊢ 𝑂 = (oppg‘𝐺) ⇒ ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂) | ||
Theorem | oppgcntz 17617 | A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝑂 = (oppg‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ (𝑍‘𝐴) = ((Cntz‘𝑂)‘𝐴) | ||
Theorem | oppgcntr 17618 | The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝑂 = (oppg‘𝐺) & ⊢ 𝑍 = (Cntr‘𝐺) ⇒ ⊢ 𝑍 = (Cntr‘𝑂) | ||
Theorem | gsumwrev 17619 | A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑂 = (oppg‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊))) | ||
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 17621. However, the
set is allowed to be empty, see symgbas0 17637. Hint: The symmetric groups
should not be confused with "symmetry groups" which is a different topic in
group theory.
| ||
Syntax | csymg 17620 | Extend class notation to include the class of symmetric groups. |
class SymGrp | ||
Definition | df-symg 17621* | 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. (Contributed by Paul Chapman, 25-Feb-2008.) |
⊢ SymGrp = (𝑥 ∈ V ↦ ⦋{ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} / 𝑏⦌{〈(Base‘ndx), 𝑏〉, 〈(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))〉}) | ||
Theorem | symgval 17622* | The value of the symmetric group function at 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) |
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} & ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) & ⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴})) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉}) | ||
Theorem | symgbas 17623* | The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.) |
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} | ||
Theorem | elsymgbas2 17624 | Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Mario Carneiro, 28-Jan-2015.) |
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) | ||
Theorem | elsymgbas 17625 | 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 17626 | 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 17627 | A permutation (element of the symmetric group) is a function of a set into itself. (Contributed by AV, 1-Jan-2019.) |
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴⟶𝐴) | ||
Theorem | symghash 17628 | 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 17629 | 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 17630 | 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 17631 | 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 | symgplusg 17632* | 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.) |
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) | ||
Theorem | symgov 17633 | 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.) |
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌)) | ||
Theorem | symgcl 17634 | 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 | symgmov1 17635* | 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 17636* | 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 17637 | 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 17638 | 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 17639 | 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 17640 | 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 17641 | The symmetric group on a pair is the symmetric group S2 consisting of the identity and the transposition. This theorem is also valid if the elements are identical: then it collapses to theorem symg1bas 17639. (Contributed by AV, 9-Dec-2018.) |
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐴 = {𝐼, 𝐽} ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊) → 𝐵 = {{〈𝐼, 𝐼〉, 〈𝐽, 𝐽〉}, {〈𝐼, 𝐽〉, 〈𝐽, 𝐼〉}}) | ||
Theorem | symgtset 17642 | 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) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺)) | ||
Theorem | symggrp 17643 | 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) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) | ||
Theorem | symgid 17644 | 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) ⇒ ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝐺)) | ||
Theorem | symginv 17645 | 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 | galactghm 17646* | 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 17647* | The converse of galactghm 17646. 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 17648 | 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 17649* | 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 17650 | 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 17651* | 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) ⇒ ⊢ (𝐴 ∈ 𝑉 → ((𝑃 ∈ Grp ∧ 𝐹 ⊆ 𝐵 ∧ (+g‘𝑃) = (𝑓 ∈ 𝐹, 𝑔 ∈ 𝐹 ↦ (𝑓 ∘ 𝑔))) → 𝐹 ∈ (SubGrp‘𝐺))) | ||
Theorem | idresperm 17652 | 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 | idressubgsymg 17653 | 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 17654 | 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 17655* | Lemma for cayley 17657. (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 17656* | Lemma for cayley 17657. (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 17657* | 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 17658* | 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 17659* | 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 17660* | 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 17661* | 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 17662* | 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 17663* | Lemma for symgextf1 17664. (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 17664* | 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 17665* | 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 17666* | 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 17667* | 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 17668* | 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 17669 | 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 17670* | Lemma 1 for gsmsymgrfix 17671. (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 17671* | 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 17672* | 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 17673* | Lemma 1 for gsmsymgreq 17675. (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 17674* | Lemma 2 for gsmsymgreq 17675. (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 17675* | 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 17676* | 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 17677* | 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 17678* | 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 17679* | 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 17680* | 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 17681* | Lemma 1 for symgfixfo 17682. (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 17682* | 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 17683* | 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 17684 | Syntax for the transposition generator function. |
class pmTrsp | ||
Definition | df-pmtr 17685* | 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 17686 | 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 17687 | 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 17688 | 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 17689 | 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 17690 | 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 17691 | 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 17692 | 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 17693* | 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 17694* | 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 17695 | 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 17696 | 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 17697 | 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 17698 | 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 17699 | 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 17700 | 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) |
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