HomeHome Metamath Proof Explorer
Theorem List (p. 308 of 449)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28687)
  Hilbert Space Explorer  Hilbert Space Explorer
(28688-30210)
  Users' Mathboxes  Users' Mathboxes
(30211-44891)
 

Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremomndaddr 30701 In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)       (((oppg𝑀) ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑍 + 𝑋) (𝑍 + 𝑌))
 
Theoremomndadd2d 30702 In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑊𝐵)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑍)    &   (𝜑𝑌 𝑊)    &   (𝜑𝑀 ∈ CMnd)       (𝜑 → (𝑋 + 𝑌) (𝑍 + 𝑊))
 
Theoremomndadd2rd 30703 In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑊𝐵)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑍)    &   (𝜑𝑌 𝑊)    &   (𝜑 → (oppg𝑀) ∈ oMnd)       (𝜑 → (𝑋 + 𝑌) (𝑍 + 𝑊))
 
Theoremsubmomnd 30704 A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)
 
Theoremxrge0omnd 30705 The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.)
(ℝ*𝑠s (0[,]+∞)) ∈ oMnd
 
Theoremomndmul2 30706 In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))
 
Theoremomndmul3 30707 In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &    0 = (0g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑃 ∈ ℕ0)    &   (𝜑𝑁𝑃)    &   (𝜑𝑋𝐵)    &   (𝜑0 𝑋)       (𝜑 → (𝑁 · 𝑋) (𝑃 · 𝑋))
 
Theoremomndmul 30708 In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑀 ∈ CMnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 𝑌)       (𝜑 → (𝑁 · 𝑋) (𝑁 · 𝑌))
 
Theoremogrpinv0le 30709 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    = (le‘𝐺)    &   𝐼 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 𝑋 ↔ (𝐼𝑋) 0 ))
 
Theoremogrpsub 30710 In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝐺)    &    = (le‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) (𝑌 𝑍))
 
Theoremogrpaddlt 30711 In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
 
Theoremogrpaddltbi 30712 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))
 
Theoremogrpaddltrd 30713 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑 → (oppg𝐺) ∈ oGrp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 < 𝑌)       (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌))
 
Theoremogrpaddltrbid 30714 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑 → (oppg𝐺) ∈ oGrp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 < 𝑌 ↔ (𝑍 + 𝑋) < (𝑍 + 𝑌)))
 
Theoremogrpsublt 30715 In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) < (𝑌 𝑍))
 
Theoremogrpinv0lt 30716 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &   𝐼 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))
 
Theoremogrpinvlt 30717 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &   𝐼 = (invg𝐺)       (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝐼𝑌) < (𝐼𝑋)))
 
Theoremgsumle 30718 A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑀 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝐹r 𝐺)       (𝜑 → (𝑀 Σg 𝐹) (𝑀 Σg 𝐺))
 
20.3.9.5  The symmetric group
 
Theoremsymgfcoeu 30719* Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐺 = (Base‘(SymGrp‘𝐷))       ((𝐷𝑉𝑃𝐺𝑄𝐺) → ∃!𝑝𝐺 𝑄 = (𝑃𝑝))
 
Theoremsymgcom 30720 Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋𝐸) = ( I ↾ 𝐸))    &   (𝜑 → (𝑌𝐹) = ( I ↾ 𝐹))    &   (𝜑 → (𝐸𝐹) = ∅)    &   (𝜑 → (𝐸𝐹) = 𝐴)       (𝜑 → (𝑋𝑌) = (𝑌𝑋))
 
Theoremsymgcom2 30721 Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 17-Nov-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅)       (𝜑 → (𝑋𝑌) = (𝑌𝑋))
 
Theoremsymgcntz 30722* All elements of a (finite) set of permutations commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (Cntz‘𝑆)    &   (𝜑𝐴𝐵)    &   (𝜑Disj 𝑥𝐴 dom (𝑥 ∖ I ))       (𝜑𝐴 ⊆ (𝑍𝐴))
 
Theoremodpmco 30723 The composition of two odd permutations is even. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (pmEven‘𝐷)       ((𝐷 ∈ Fin ∧ 𝑋 ∈ (𝐵𝐴) ∧ 𝑌 ∈ (𝐵𝐴)) → (𝑋𝑌) ∈ 𝐴)
 
Theoremsymgsubg 30724 The value of the group subtraction operation of the symmetric group. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋𝑌))
 
Theorempmtrprfv2 30725 In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋)
 
Theorempmtrcnel 30726 Composing a permutation 𝐹 with a transposition which results in moving at least one less point. Here the set of points moved by a permutation 𝐹 is expressed as dom (𝐹 ∖ I ). (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))       (𝜑 → dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∖ {𝐼}))
 
Theorempmtrcnel2 30727 Variation on pmtrcnel 30726. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))       (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))
 
Theorempmtrcnelor 30728 Composing a permutation 𝐹 with a transposition which results in moving one or two less points. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))    &   𝐸 = dom (𝐹 ∖ I )    &   𝐴 = dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )       (𝜑 → (𝐴 = (𝐸 ∖ {𝐼, 𝐽}) ∨ 𝐴 = (𝐸 ∖ {𝐼})))
 
20.3.9.6  Transpositions
 
Theorempmtridf1o 30729 Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑𝑇:𝐴1-1-onto𝐴)
 
Theorempmtridfv1 30730 Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑋) = 𝑌)
 
Theorempmtridfv2 30731 Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑌) = 𝑋)
 
20.3.9.7  Permutation Signs
 
Theorempsgnid 30732 Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝑆 = (pmSgn‘𝐷)       (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)
 
Theorempsgndmfi 30733 For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑆 = (pmSgn‘𝐷)    &   𝐺 = (Base‘(SymGrp‘𝐷))       (𝐷 ∈ Fin → 𝑆 Fn 𝐺)
 
Theorempmtrto1cl 30734 Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑇 = (pmTrsp‘𝐷)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇)
 
Theorempsgnfzto1stlem 30735* Lemma for psgnfzto1st 30740. Our permutation of rank (𝑛 + 1) can be written as a permutation of rank 𝑛 composed with a transposition. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≤ (𝐾 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝐾, (𝐾 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖𝐾, (𝑖 − 1), 𝑖)))))
 
Theoremfzto1stfv1 30736* Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼𝐷 → (𝑃‘1) = 𝐼)
 
Theoremfzto1st1 30737* Special case where the permutation defined in psgnfzto1st 30740 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼 = 1 → 𝑃 = ( I ↾ 𝐷))
 
Theoremfzto1st 30738* The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷𝑃𝐵)
 
Theoremfzto1stinvn 30739* Value of the inverse of our permutation 𝑃 at 𝐼 (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷 → (𝑃𝐼) = 1)
 
Theorempsgnfzto1st 30740* The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑆 = (pmSgn‘𝐷)       (𝐼𝐷 → (𝑆𝑃) = (-1↑(𝐼 + 1)))
 
20.3.9.8  Permutation cycles
 
Syntaxctocyc 30741 Extend class notation with the permutation cycle builder.
class toCyc
 
Definitiondf-tocyc 30742* Define a convenience permutation cycle builder. Given a list of elements to be cycled, in the form of a word, this function produces the corresponding permutation cycle. See definition in [Lang] p. 30. (Contributed by Thierry Arnoux, 19-Sep-2023.)
toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑𝑢:dom 𝑢1-1𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ 𝑤))))
 
Theoremtocycval 30743* Value of the cycle builder. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)       (𝐷𝑉𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷𝑢:dom 𝑢1-1𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ 𝑤))))
 
Theoremtocycfv 30744 Function value of a permutation cycle built from a word. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → (𝐶𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊)))
 
Theoremtocycfvres1 30745 A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → ((𝐶𝑊) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ 𝑊))
 
Theoremtocycfvres2 30746 A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → ((𝐶𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊)))
 
Theoremcycpmfvlem 30747 Lemma for cycpmfv1 30748 and cycpmfv2 30749. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑁 ∈ (0..^(♯‘𝑊)))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (((𝑊 cyclShift 1) ∘ 𝑊)‘(𝑊𝑁)))
 
Theoremcycpmfv1 30748 Value of a cycle function for any element but the last. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑁 ∈ (0..^((♯‘𝑊) − 1)))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (𝑊‘(𝑁 + 1)))
 
Theoremcycpmfv2 30749 Value of a cycle function for the last element of the orbit. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → 0 < (♯‘𝑊))    &   (𝜑𝑁 = ((♯‘𝑊) − 1))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (𝑊‘0))
 
Theoremcycpmfv3 30750 Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑋𝐷)    &   (𝜑 → ¬ 𝑋 ∈ ran 𝑊)       (𝜑 → ((𝐶𝑊)‘𝑋) = 𝑋)
 
Theoremcycpmcl 30751 Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → (𝐶𝑊) ∈ (Base‘𝑆))
 
Theoremtocycf 30752* The permutation cycle builder as a function. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)       (𝐷𝑉𝐶:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶𝐵)
 
Theoremtocyc01 30753 Permutation cycles built from the empty set or a singleton are the identity. (Contributed by Thierry Arnoux, 21-Nov-2023.)
𝐶 = (toCyc‘𝐷)       ((𝐷𝑉𝑊 ∈ (dom 𝐶 ∩ (♯ “ {0, 1}))) → (𝐶𝑊) = ( I ↾ 𝐷))
 
Theoremcycpm2tr 30754 A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑇 = (pmTrsp‘𝐷)       (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) = (𝑇‘{𝐼, 𝐽}))
 
Theoremcycpm2cl 30755 Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
 
Theoremcyc2fv1 30756 Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
 
Theoremcyc2fv2 30757 Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
 
Theoremtrsp2cyc 30758* Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐶 = (toCyc‘𝐷)       ((𝐷𝑉𝑃𝑇) → ∃𝑖𝐷𝑗𝐷 (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
 
Theoremcycpmco2f1 30759 The word U used in cycpmco2 30768 is injective, so it can represent a cycle and form a cyclic permutation (𝑀𝑈). (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑𝑈:dom 𝑈1-1𝐷)
 
Theoremcycpmco2rn 30760 The orbit of the composition of a cyclic permutation and a well-chosen transposition is one element more than the orbit of the original permutation. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
 
Theoremcycpmco2lem1 30761 Lemma for cycpmco2 30768. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑊)‘𝐽))
 
Theoremcycpmco2lem2 30762 Lemma for cycpmco2 30768. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → (𝑈𝐸) = 𝐼)
 
Theoremcycpmco2lem3 30763 Lemma for cycpmco2 30768. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
 
Theoremcycpmco2lem4 30764 Lemma for cycpmco2 30768. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑈)‘𝐼))
 
Theoremcycpmco2lem5 30765 Lemma for cycpmco2 30768. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑 → (𝑈𝐾) = ((♯‘𝑈) − 1))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
Theoremcycpmco2lem6 30766 Lemma for cycpmco2 30768. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑𝐾𝐼)    &   (𝜑 → (𝑈𝐾) ∈ (𝐸..^((♯‘𝑈) − 1)))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
Theoremcycpmco2lem7 30767 Lemma for cycpmco2 30768. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑𝐾𝐽)    &   (𝜑 → (𝑈𝐾) ∈ (0..^𝐸))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
Theoremcycpmco2 30768 The composition of a cyclic permutation and a transposition of one element in the cycle and one outside the cycle results in a cyclic permutation with one more element in its orbit. (Contributed by Thierry Arnoux, 2-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀𝑈))
 
Theoremcyc2fvx 30769 Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
 
Theoremcycpm3cl 30770 Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
 
Theoremcycpm3cl2 30771 Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (𝐶 “ (♯ “ {3})))
 
Theoremcyc3fv1 30772 Function value of a 3-cycle at the first point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
 
Theoremcyc3fv2 30773 Function value of a 3-cycle at the second point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
 
Theoremcyc3fv3 30774 Function value of a 3-cycle at the third point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
 
Theoremcyc3co2 30775 Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)    &    · = (+g𝑆)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))
 
Theoremcycpmconjvlem 30776 Lemma for cycpmconjv 30777 (Contributed by Thierry Arnoux, 9-Oct-2023.)
(𝜑𝐹:𝐷1-1-onto𝐷)    &   (𝜑𝐵𝐷)       (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
 
Theoremcycpmconjv 30777 A formula for computing conjugacy classes of cyclic permutations. Formula in property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 9-Oct-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    + = (+g𝑆)    &    = (-g𝑆)    &   𝐵 = (Base‘𝑆)       ((𝐷𝑉𝐺𝐵𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀𝑊)) 𝐺) = (𝑀‘(𝐺𝑊)))
 
Theoremcycpmrn 30778 The range of the word used to build a cycle is the cycle's orbit, i.e. the set of points it moves. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → 1 < (♯‘𝑊))       (𝜑 → ran 𝑊 = dom ((𝑀𝑊) ∖ I ))
 
Theoremtocyccntz 30779* All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑍 = (Cntz‘𝑆)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑Disj 𝑥𝐴 ran 𝑥)    &   (𝜑𝐴 ⊆ dom 𝑀)       (𝜑 → (𝑀𝐴) ⊆ (𝑍‘(𝑀𝐴)))
 
20.3.9.9  The Alternating Group
 
Theoremevpmval 30780 Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝐴 = (pmEven‘𝐷)       (𝐷𝑉𝐴 = ((pmSgn‘𝐷) “ {1}))
 
Theoremcnmsgn0g 30781 The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})       1 = (0g𝑈)
 
Theoremevpmsubg 30782 The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐴 = (pmEven‘𝐷)       (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆))
 
Theoremevpmid 30783 The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrp‘𝐷)       (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷))
 
Theoremaltgnsg 30784 The alternating group (pmEven‘𝐷) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrp‘𝐷)       (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
 
Theoremcyc3evpm 30785 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = ((toCyc‘𝐷) “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)       (𝐷 ∈ Fin → 𝐶𝐴)
 
Theoremcyc3genpmlem 30786* Lemma for cyc3genpm 30787. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    · = (+g𝑆)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐿𝐷)    &   (𝜑𝐸 = (𝑀‘⟨“𝐼𝐽”⟩))    &   (𝜑𝐹 = (𝑀‘⟨“𝐾𝐿”⟩))    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐽)    &   (𝜑𝐾𝐿)       (𝜑 → ∃𝑐 ∈ Word 𝐶(𝐸 · 𝐹) = (𝑆 Σg 𝑐))
 
Theoremcyc3genpm 30787* The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)       (𝐷 ∈ Fin → (𝑄𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))
 
Theoremcycpmgcl 30788 Cyclic permutations are permutations, similar to cycpmcl 30751, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)       ((𝐷𝑉𝑃 ∈ (0...𝑁)) → 𝐶𝐵)
 
Theoremcycpmconjslem1 30789 Lemma for cycpmconjs 30791 (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → (♯‘𝑊) = 𝑃)       (𝜑 → ((𝑊 ∘ (𝑀𝑊)) ∘ 𝑊) = (( I ↾ (0..^𝑃)) cyclShift 1))
 
Theoremcycpmconjslem2 30790* Lemma for cycpmconjs 30791 (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑𝑃 ∈ (0...𝑁))    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)       (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
 
Theoremcycpmconjs 30791* All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑𝑃 ∈ (0...𝑁))    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)    &   (𝜑𝑇𝐶)       (𝜑 → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
 
Theoremcyc3conja 30792* All 3-cycles are conjugate in the alternating group An for n>= 5. Property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑 → 5 ≤ 𝑁)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)    &   (𝜑𝑇𝐶)       (𝜑 → ∃𝑝𝐴 𝑄 = ((𝑝 + 𝑇) 𝑝))
 
20.3.9.10  Signum in an ordered monoid
 
Syntaxcsgns 30793 Extend class notation to include the Signum function.
class sgns
 
Definitiondf-sgns 30794* Signum function for a structure. See also df-sgn 14438 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.)
sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
 
Theoremsgnsv 30795* The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
 
Theoremsgnsval 30796 The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       ((𝑅𝑉𝑋𝐵) → (𝑆𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
 
Theoremsgnsf 30797 The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆:𝐵⟶{-1, 0, 1})
 
20.3.9.11  The Archimedean property for generic ordered algebraic structures
 
Syntaxcinftm 30798 Class notation for the infinitesimal relation.
class
 
Syntaxcarchi 30799 Class notation for the Archimedean property.
class Archi
 
Definitiondf-inftm 30800* Define the relation "𝑥 is infinitesimal with respect to 𝑦 " for a structure 𝑤. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44891
  Copyright terms: Public domain < Previous  Next >