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Theorem List for Metamath Proof Explorer - 32101-32200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremomndmul2 32101 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 32102 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 32103 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 32104 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 32105 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 32106 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 32107 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 32108 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 32109 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 32110 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 32111 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 32112 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 32113 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))
 
21.3.9.5  The symmetric group
 
Theoremsymgfcoeu 32114* Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐺 = (Base‘(SymGrp‘𝐷))       ((𝐷𝑉𝑃𝐺𝑄𝐺) → ∃!𝑝𝐺 𝑄 = (𝑃𝑝))
 
Theoremsymgcom 32115 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 32116 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 32117* 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 32118 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 32119 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 32120 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 32121 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 32122 Variation on pmtrcnel 32121. (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 32123 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})))
 
21.3.9.6  Transpositions
 
Theorempmtridf1o 32124 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 32125 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 32126 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)
 
21.3.9.7  Permutation Signs
 
Theorempsgnid 32127 Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝑆 = (pmSgn‘𝐷)       (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)
 
Theorempsgndmfi 32128 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 32129 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 32130* Lemma for psgnfzto1st 32135. 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 32131* 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 32132* Special case where the permutation defined in psgnfzto1st 32135 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 32133* 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 32134* 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 32135* 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)))
 
21.3.9.8  Permutation cycles
 
Syntaxctocyc 32136 Extend class notation with the permutation cycle builder.
class toCyc
 
Definitiondf-tocyc 32137* 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 32138* 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 32139 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 32140 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 32141 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 32142 Lemma for cycpmfv1 32143 and cycpmfv2 32144. (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 32143 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 32144 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 32145 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 32146 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 32147* 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 32148 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 32149 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 32150 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 32151 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 32152 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 32153* 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 32154 The word U used in cycpmco2 32163 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 32155 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 32156 Lemma for cycpmco2 32163. (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 32157 Lemma for cycpmco2 32163. (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 32158 Lemma for cycpmco2 32163. (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 32159 Lemma for cycpmco2 32163. (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 32160 Lemma for cycpmco2 32163. (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 32161 Lemma for cycpmco2 32163. (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 32162 Lemma for cycpmco2 32163. (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 32163 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 32164 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 32165 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 32166 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 32167 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 32168 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 32169 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 32170 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 32171 Lemma for cycpmconjv 32172. (Contributed by Thierry Arnoux, 9-Oct-2023.)
(𝜑𝐹:𝐷1-1-onto𝐷)    &   (𝜑𝐵𝐷)       (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
 
Theoremcycpmconjv 32172 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 32173 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 32174* 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)))
 
21.3.9.9  The Alternating Group
 
Theoremevpmval 32175 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 32176 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 32177 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 32178 The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrp‘𝐷)       (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷))
 
Theoremaltgnsg 32179 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 32180 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = ((toCyc‘𝐷) “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)       (𝐷 ∈ Fin → 𝐶𝐴)
 
Theoremcyc3genpmlem 32181* Lemma for cyc3genpm 32182. (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 32182* 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 32183 Cyclic permutations are permutations, similar to cycpmcl 32146, 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 32184 Lemma for cycpmconjs 32186. (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 32185* Lemma for cycpmconjs 32186. (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 32186* 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 32187* 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))
 
21.3.9.10  Signum in an ordered monoid
 
Syntaxcsgns 32188 Extend class notation to include the Signum function.
class sgns
 
Definitiondf-sgns 32189* Signum function for a structure. See also df-sgn 15016 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 32190* 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 32191 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 32192 The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆:𝐵⟶{-1, 0, 1})
 
21.3.9.11  The Archimedean property for generic ordered algebraic structures
 
Syntaxcinftm 32193 Class notation for the infinitesimal relation.
class
 
Syntaxcarchi 32194 Class notation for the Archimedean property.
class Archi
 
Definitiondf-inftm 32195* 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))})
 
Definitiondf-archi 32196 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
 
Theoreminftmrel 32197 The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)       (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
 
Theoremisinftm 32198* Express 𝑥 is infinitesimal with respect to 𝑦 for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    · = (.g𝑊)    &    < = (lt‘𝑊)       ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
 
Theoremisarchi 32199* Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (⋘‘𝑊)       (𝑊𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
 
Theorempnfinf 32200 Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018.)
(𝐴 ∈ ℝ+𝐴(⋘‘ℝ*𝑠)+∞)
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