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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))})

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