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Theorem List for Metamath Proof Explorer - 31701-31800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrge0omnd 31701 The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.)
(ℝ*𝑠 β†Ύs (0[,]+∞)) ∈ oMnd
 
Theoremomndmul2 31702 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.)
𝐡 = (Baseβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    &    Β· = (.gβ€˜π‘€)    &    0 = (0gβ€˜π‘€)    β‡’   ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•0) ∧ 0 ≀ 𝑋) β†’ 0 ≀ (𝑁 Β· 𝑋))
 
Theoremomndmul3 31703 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.)
𝐡 = (Baseβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    &    Β· = (.gβ€˜π‘€)    &    0 = (0gβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ oMnd)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    &   (πœ‘ β†’ 𝑁 ≀ 𝑃)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 0 ≀ 𝑋)    β‡’   (πœ‘ β†’ (𝑁 Β· 𝑋) ≀ (𝑃 Β· 𝑋))
 
Theoremomndmul 31704 In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    &    Β· = (.gβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ oMnd)    &   (πœ‘ β†’ 𝑀 ∈ CMnd)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    β‡’   (πœ‘ β†’ (𝑁 Β· 𝑋) ≀ (𝑁 Β· π‘Œ))
 
Theoremogrpinv0le 31705 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐡 = (Baseβ€˜πΊ)    &    ≀ = (leβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) β†’ ( 0 ≀ 𝑋 ↔ (πΌβ€˜π‘‹) ≀ 0 ))
 
Theoremogrpsub 31706 In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜πΊ)    &    ≀ = (leβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 βˆ’ 𝑍) ≀ (π‘Œ βˆ’ 𝑍))
 
Theoremogrpaddlt 31707 In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐡 = (Baseβ€˜πΊ)    &    < = (ltβ€˜πΊ)    &    + = (+gβ€˜πΊ)    β‡’   ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ (𝑋 + 𝑍) < (π‘Œ + 𝑍))
 
Theoremogrpaddltbi 31708 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐡 = (Baseβ€˜πΊ)    &    < = (ltβ€˜πΊ)    &    + = (+gβ€˜πΊ)    β‡’   ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 < π‘Œ ↔ (𝑋 + 𝑍) < (π‘Œ + 𝑍)))
 
Theoremogrpaddltrd 31709 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐡 = (Baseβ€˜πΊ)    &    < = (ltβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    &   (πœ‘ β†’ (oppgβ€˜πΊ) ∈ oGrp)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 < π‘Œ)    β‡’   (πœ‘ β†’ (𝑍 + 𝑋) < (𝑍 + π‘Œ))
 
Theoremogrpaddltrbid 31710 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝐡 = (Baseβ€˜πΊ)    &    < = (ltβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    &   (πœ‘ β†’ (oppgβ€˜πΊ) ∈ oGrp)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 < π‘Œ ↔ (𝑍 + 𝑋) < (𝑍 + π‘Œ)))
 
Theoremogrpsublt 31711 In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐡 = (Baseβ€˜πΊ)    &    < = (ltβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ (𝑋 βˆ’ 𝑍) < (π‘Œ βˆ’ 𝑍))
 
Theoremogrpinv0lt 31712 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐡 = (Baseβ€˜πΊ)    &    < = (ltβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐡) β†’ ( 0 < 𝑋 ↔ (πΌβ€˜π‘‹) < 0 ))
 
Theoremogrpinvlt 31713 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐡 = (Baseβ€˜πΊ)    &    < = (ltβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    β‡’   (((𝐺 ∈ oGrp ∧ (oppgβ€˜πΊ) ∈ oGrp) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ ↔ (πΌβ€˜π‘Œ) < (πΌβ€˜π‘‹)))
 
Theoremgsumle 31714 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.)
𝐡 = (Baseβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ oMnd)    &   (πœ‘ β†’ 𝑀 ∈ CMnd)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝐺:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝐹 ∘r ≀ 𝐺)    β‡’   (πœ‘ β†’ (𝑀 Ξ£g 𝐹) ≀ (𝑀 Ξ£g 𝐺))
 
21.3.9.5  The symmetric group
 
Theoremsymgfcoeu 31715* Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐺 = (Baseβ€˜(SymGrpβ€˜π·))    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝐺 ∧ 𝑄 ∈ 𝐺) β†’ βˆƒ!𝑝 ∈ 𝐺 𝑄 = (𝑃 ∘ 𝑝))
 
Theoremsymgcom 31716 Two permutations 𝑋 and π‘Œ commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrpβ€˜π΄)    &   π΅ = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ (𝑋 β†Ύ 𝐸) = ( I β†Ύ 𝐸))    &   (πœ‘ β†’ (π‘Œ β†Ύ 𝐹) = ( I β†Ύ 𝐹))    &   (πœ‘ β†’ (𝐸 ∩ 𝐹) = βˆ…)    &   (πœ‘ β†’ (𝐸 βˆͺ 𝐹) = 𝐴)    β‡’   (πœ‘ β†’ (𝑋 ∘ π‘Œ) = (π‘Œ ∘ 𝑋))
 
Theoremsymgcom2 31717 Two permutations 𝑋 and π‘Œ commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 17-Nov-2023.)
𝐺 = (SymGrpβ€˜π΄)    &   π΅ = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ (dom (𝑋 βˆ– I ) ∩ dom (π‘Œ βˆ– I )) = βˆ…)    β‡’   (πœ‘ β†’ (𝑋 ∘ π‘Œ) = (π‘Œ ∘ 𝑋))
 
Theoremsymgcntz 31718* All elements of a (finite) set of permutations commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &   π‘ = (Cntzβ€˜π‘†)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   (πœ‘ β†’ Disj π‘₯ ∈ 𝐴 dom (π‘₯ βˆ– I ))    β‡’   (πœ‘ β†’ 𝐴 βŠ† (π‘β€˜π΄))
 
Theoremodpmco 31719 The composition of two odd permutations is even. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &   π΄ = (pmEvenβ€˜π·)    β‡’   ((𝐷 ∈ Fin ∧ 𝑋 ∈ (𝐡 βˆ– 𝐴) ∧ π‘Œ ∈ (𝐡 βˆ– 𝐴)) β†’ (𝑋 ∘ π‘Œ) ∈ 𝐴)
 
Theoremsymgsubg 31720 The value of the group subtraction operation of the symmetric group. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrpβ€˜π΄)    &   π΅ = (Baseβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 βˆ’ π‘Œ) = (𝑋 ∘ β—‘π‘Œ))
 
Theorempmtrprfv2 31721 In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑇 = (pmTrspβ€˜π·)    β‡’   ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ π‘Œ ∈ 𝐷 ∧ 𝑋 β‰  π‘Œ)) β†’ ((π‘‡β€˜{𝑋, π‘Œ})β€˜π‘Œ) = 𝑋)
 
Theorempmtrcnel 31722 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 βˆ– I ). (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π‘‡ = (pmTrspβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &   π½ = (πΉβ€˜πΌ)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐼 ∈ dom (𝐹 βˆ– I ))    β‡’   (πœ‘ β†’ dom (((π‘‡β€˜{𝐼, 𝐽}) ∘ 𝐹) βˆ– I ) βŠ† (dom (𝐹 βˆ– I ) βˆ– {𝐼}))
 
Theorempmtrcnel2 31723 Variation on pmtrcnel 31722. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π‘‡ = (pmTrspβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &   π½ = (πΉβ€˜πΌ)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐼 ∈ dom (𝐹 βˆ– I ))    β‡’   (πœ‘ β†’ (dom (𝐹 βˆ– I ) βˆ– {𝐼, 𝐽}) βŠ† dom (((π‘‡β€˜{𝐼, 𝐽}) ∘ 𝐹) βˆ– I ))
 
Theorempmtrcnelor 31724 Composing a permutation 𝐹 with a transposition which results in moving one or two less points. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π‘‡ = (pmTrspβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &   π½ = (πΉβ€˜πΌ)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐼 ∈ dom (𝐹 βˆ– I ))    &   πΈ = dom (𝐹 βˆ– I )    &   π΄ = dom (((π‘‡β€˜{𝐼, 𝐽}) ∘ 𝐹) βˆ– I )    β‡’   (πœ‘ β†’ (𝐴 = (𝐸 βˆ– {𝐼, 𝐽}) ∨ 𝐴 = (𝐸 βˆ– {𝐼})))
 
21.3.9.6  Transpositions
 
Theorempmtridf1o 31725 Transpositions of 𝑋 and π‘Œ (understood to be the identity when 𝑋 = π‘Œ), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘Œ ∈ 𝐴)    &   π‘‡ = if(𝑋 = π‘Œ, ( I β†Ύ 𝐴), ((pmTrspβ€˜π΄)β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ 𝑇:𝐴–1-1-onto→𝐴)
 
Theorempmtridfv1 31726 Value at X of the transposition of 𝑋 and π‘Œ (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘Œ ∈ 𝐴)    &   π‘‡ = if(𝑋 = π‘Œ, ( I β†Ύ 𝐴), ((pmTrspβ€˜π΄)β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ (π‘‡β€˜π‘‹) = π‘Œ)
 
Theorempmtridfv2 31727 Value at Y of the transposition of 𝑋 and π‘Œ (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘Œ ∈ 𝐴)    &   π‘‡ = if(𝑋 = π‘Œ, ( I β†Ύ 𝐴), ((pmTrspβ€˜π΄)β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ (π‘‡β€˜π‘Œ) = 𝑋)
 
21.3.9.7  Permutation Signs
 
Theorempsgnid 31728 Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝑆 = (pmSgnβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ (π‘†β€˜( I β†Ύ 𝐷)) = 1)
 
Theorempsgndmfi 31729 For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑆 = (pmSgnβ€˜π·)    &   πΊ = (Baseβ€˜(SymGrpβ€˜π·))    β‡’   (𝐷 ∈ Fin β†’ 𝑆 Fn 𝐺)
 
Theorempmtrto1cl 31730 Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘‡ = (pmTrspβ€˜π·)    β‡’   ((𝐾 ∈ β„• ∧ (𝐾 + 1) ∈ 𝐷) β†’ (π‘‡β€˜{𝐾, (𝐾 + 1)}) ∈ ran 𝑇)
 
Theorempsgnfzto1stlem 31731* Lemma for psgnfzto1st 31736. 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)    β‡’   ((𝐾 ∈ β„• ∧ (𝐾 + 1) ∈ 𝐷) β†’ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≀ (𝐾 + 1), (𝑖 βˆ’ 1), 𝑖))) = (((pmTrspβ€˜π·)β€˜{𝐾, (𝐾 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖 ≀ 𝐾, (𝑖 βˆ’ 1), 𝑖)))))
 
Theoremfzto1stfv1 31732* Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    β‡’   (𝐼 ∈ 𝐷 β†’ (π‘ƒβ€˜1) = 𝐼)
 
Theoremfzto1st1 31733* Special case where the permutation defined in psgnfzto1st 31736 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    β‡’   (𝐼 = 1 β†’ 𝑃 = ( I β†Ύ 𝐷))
 
Theoremfzto1st 31734* 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)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   πΊ = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝐼 ∈ 𝐷 β†’ 𝑃 ∈ 𝐡)
 
Theoremfzto1stinvn 31735* Value of the inverse of our permutation 𝑃 at 𝐼. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   πΊ = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝐼 ∈ 𝐷 β†’ (β—‘π‘ƒβ€˜πΌ) = 1)
 
Theorempsgnfzto1st 31736* The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   πΊ = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜πΊ)    &   π‘† = (pmSgnβ€˜π·)    β‡’   (𝐼 ∈ 𝐷 β†’ (π‘†β€˜π‘ƒ) = (-1↑(𝐼 + 1)))
 
21.3.9.8  Permutation cycles
 
Syntaxctocyc 31737 Extend class notation with the permutation cycle builder.
class toCyc
 
Definitiondf-tocyc 31738* 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 ↦ (𝑀 ∈ {𝑒 ∈ Word 𝑑 ∣ 𝑒:dom 𝑒–1-1→𝑑} ↦ (( I β†Ύ (𝑑 βˆ– ran 𝑀)) βˆͺ ((𝑀 cyclShift 1) ∘ ◑𝑀))))
 
Theoremtocycval 31739* Value of the cycle builder. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    β‡’   (𝐷 ∈ 𝑉 β†’ 𝐢 = (𝑀 ∈ {𝑒 ∈ Word 𝐷 ∣ 𝑒:dom 𝑒–1-1→𝐷} ↦ (( I β†Ύ (𝐷 βˆ– ran 𝑀)) βˆͺ ((𝑀 cyclShift 1) ∘ ◑𝑀))))
 
Theoremtocycfv 31740 Function value of a permutation cycle built from a word. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    β‡’   (πœ‘ β†’ (πΆβ€˜π‘Š) = (( I β†Ύ (𝐷 βˆ– ran π‘Š)) βˆͺ ((π‘Š cyclShift 1) ∘ β—‘π‘Š)))
 
Theoremtocycfvres1 31741 A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š) β†Ύ ran π‘Š) = ((π‘Š cyclShift 1) ∘ β—‘π‘Š))
 
Theoremtocycfvres2 31742 A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š) β†Ύ (𝐷 βˆ– ran π‘Š)) = ( I β†Ύ (𝐷 βˆ– ran π‘Š)))
 
Theoremcycpmfvlem 31743 Lemma for cycpmfv1 31744 and cycpmfv2 31745. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 𝑁 ∈ (0..^(β™―β€˜π‘Š)))    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š)β€˜(π‘Šβ€˜π‘)) = (((π‘Š cyclShift 1) ∘ β—‘π‘Š)β€˜(π‘Šβ€˜π‘)))
 
Theoremcycpmfv1 31744 Value of a cycle function for any element but the last. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 𝑁 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)))    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š)β€˜(π‘Šβ€˜π‘)) = (π‘Šβ€˜(𝑁 + 1)))
 
Theoremcycpmfv2 31745 Value of a cycle function for the last element of the orbit. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 0 < (β™―β€˜π‘Š))    &   (πœ‘ β†’ 𝑁 = ((β™―β€˜π‘Š) βˆ’ 1))    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š)β€˜(π‘Šβ€˜π‘)) = (π‘Šβ€˜0))
 
Theoremcycpmfv3 31746 Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 𝑋 ∈ 𝐷)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ ran π‘Š)    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š)β€˜π‘‹) = 𝑋)
 
Theoremcycpmcl 31747 Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   π‘† = (SymGrpβ€˜π·)    β‡’   (πœ‘ β†’ (πΆβ€˜π‘Š) ∈ (Baseβ€˜π‘†))
 
Theoremtocycf 31748* The permutation cycle builder as a function. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    β‡’   (𝐷 ∈ 𝑉 β†’ 𝐢:{𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷}⟢𝐡)
 
Theoremtocyc01 31749 Permutation cycles built from the empty set or a singleton are the identity. (Contributed by Thierry Arnoux, 21-Nov-2023.)
𝐢 = (toCycβ€˜π·)    β‡’   ((𝐷 ∈ 𝑉 ∧ π‘Š ∈ (dom 𝐢 ∩ (β—‘β™― β€œ {0, 1}))) β†’ (πΆβ€˜π‘Š) = ( I β†Ύ 𝐷))
 
Theoremcycpm2tr 31750 A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   π‘‡ = (pmTrspβ€˜π·)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©) = (π‘‡β€˜{𝐼, 𝐽}))
 
Theoremcycpm2cl 31751 Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   π‘† = (SymGrpβ€˜π·)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©) ∈ (Baseβ€˜π‘†))
 
Theoremcyc2fv1 31752 Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   π‘† = (SymGrpβ€˜π·)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜πΌ) = 𝐽)
 
Theoremcyc2fv2 31753 Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   π‘† = (SymGrpβ€˜π·)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜π½) = 𝐼)
 
Theoremtrsp2cyc 31754* Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝑇 = ran (pmTrspβ€˜π·)    &   πΆ = (toCycβ€˜π·)    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) β†’ βˆƒπ‘– ∈ 𝐷 βˆƒπ‘— ∈ 𝐷 (𝑖 β‰  𝑗 ∧ 𝑃 = (πΆβ€˜βŸ¨β€œπ‘–π‘—β€βŸ©)))
 
Theoremcycpmco2f1 31755 The word U used in cycpmco2 31764 is injective, so it can represent a cycle and form a cyclic permutation (π‘€β€˜π‘ˆ). (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ π‘ˆ:dom π‘ˆβ€“1-1→𝐷)
 
Theoremcycpmco2rn 31756 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 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ran π‘ˆ = (ran π‘Š βˆͺ {𝐼}))
 
Theoremcycpmco2lem1 31757 Lemma for cycpmco2 31764. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘Š)β€˜((π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜πΌ)) = ((π‘€β€˜π‘Š)β€˜π½))
 
Theoremcycpmco2lem2 31758 Lemma for cycpmco2 31764. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ (π‘ˆβ€˜πΈ) = 𝐼)
 
Theoremcycpmco2lem3 31759 Lemma for cycpmco2 31764. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ((β™―β€˜π‘ˆ) βˆ’ 1) = (β™―β€˜π‘Š))
 
Theoremcycpmco2lem4 31760 Lemma for cycpmco2 31764. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘Š)β€˜((π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜πΌ)) = ((π‘€β€˜π‘ˆ)β€˜πΌ))
 
Theoremcycpmco2lem5 31761 Lemma for cycpmco2 31764. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    &   (πœ‘ β†’ 𝐾 ∈ ran π‘Š)    &   (πœ‘ β†’ (β—‘π‘ˆβ€˜πΎ) = ((β™―β€˜π‘ˆ) βˆ’ 1))    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘ˆ)β€˜πΎ) = ((π‘€β€˜π‘Š)β€˜πΎ))
 
Theoremcycpmco2lem6 31762 Lemma for cycpmco2 31764. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    &   (πœ‘ β†’ 𝐾 ∈ ran π‘Š)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    &   (πœ‘ β†’ (β—‘π‘ˆβ€˜πΎ) ∈ (𝐸..^((β™―β€˜π‘ˆ) βˆ’ 1)))    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘ˆ)β€˜πΎ) = ((π‘€β€˜π‘Š)β€˜πΎ))
 
Theoremcycpmco2lem7 31763 Lemma for cycpmco2 31764. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    &   (πœ‘ β†’ 𝐾 ∈ ran π‘Š)    &   (πœ‘ β†’ 𝐾 β‰  𝐽)    &   (πœ‘ β†’ (β—‘π‘ˆβ€˜πΎ) ∈ (0..^𝐸))    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘ˆ)β€˜πΎ) = ((π‘€β€˜π‘Š)β€˜πΎ))
 
Theoremcycpmco2 31764 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 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘Š) ∘ (π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©)) = (π‘€β€˜π‘ˆ))
 
Theoremcyc2fvx 31765 Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜πΎ) = 𝐾)
 
Theoremcycpm3cl 31766 Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©) ∈ (Baseβ€˜π‘†))
 
Theoremcycpm3cl2 31767 Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©) ∈ (𝐢 β€œ (β—‘β™― β€œ {3})))
 
Theoremcyc3fv1 31768 Function value of a 3-cycle at the first point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©)β€˜πΌ) = 𝐽)
 
Theoremcyc3fv2 31769 Function value of a 3-cycle at the second point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©)β€˜π½) = 𝐾)
 
Theoremcyc3fv3 31770 Function value of a 3-cycle at the third point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©)β€˜πΎ) = 𝐼)
 
Theoremcyc3co2 31771 Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    &    Β· = (+gβ€˜π‘†)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©) = ((πΆβ€˜βŸ¨β€œπΌπΎβ€βŸ©) Β· (πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)))
 
Theoremcycpmconjvlem 31772 Lemma for cycpmconjv 31773. (Contributed by Thierry Arnoux, 9-Oct-2023.)
(πœ‘ β†’ 𝐹:𝐷–1-1-onto→𝐷)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐷)    β‡’   (πœ‘ β†’ ((𝐹 β†Ύ (𝐷 βˆ– 𝐡)) ∘ ◑𝐹) = ( I β†Ύ (𝐷 βˆ– ran (𝐹 β†Ύ 𝐡))))
 
Theoremcycpmconjv 31773 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 = (SymGrpβ€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘†)    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐡 ∧ π‘Š ∈ dom 𝑀) β†’ ((𝐺 + (π‘€β€˜π‘Š)) βˆ’ 𝐺) = (π‘€β€˜(𝐺 ∘ π‘Š)))
 
Theoremcycpmrn 31774 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 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 1 < (β™―β€˜π‘Š))    β‡’   (πœ‘ β†’ ran π‘Š = dom ((π‘€β€˜π‘Š) βˆ– I ))
 
Theoremtocyccntz 31775* All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π‘ = (Cntzβ€˜π‘†)    &   π‘€ = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ Disj π‘₯ ∈ 𝐴 ran π‘₯)    &   (πœ‘ β†’ 𝐴 βŠ† dom 𝑀)    β‡’   (πœ‘ β†’ (𝑀 β€œ 𝐴) βŠ† (π‘β€˜(𝑀 β€œ 𝐴)))
 
21.3.9.9  The Alternating Group
 
Theoremevpmval 31776 Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝐴 = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ 𝑉 β†’ 𝐴 = (β—‘(pmSgnβ€˜π·) β€œ {1}))
 
Theoremcnmsgn0g 31777 The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.)
π‘ˆ = ((mulGrpβ€˜β„‚fld) β†Ύs {1, -1})    β‡’   1 = (0gβ€˜π‘ˆ)
 
Theoremevpmsubg 31778 The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π΄ = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ 𝐴 ∈ (SubGrpβ€˜π‘†))
 
Theoremevpmid 31779 The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrpβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ ( I β†Ύ 𝐷) ∈ (pmEvenβ€˜π·))
 
Theoremaltgnsg 31780 The alternating group (pmEvenβ€˜π·) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrpβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ (pmEvenβ€˜π·) ∈ (NrmSGrpβ€˜π‘†))
 
Theoremcyc3evpm 31781 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = ((toCycβ€˜π·) β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ 𝐢 βŠ† 𝐴)
 
Theoremcyc3genpmlem 31782* Lemma for cyc3genpm 31783. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    Β· = (+gβ€˜π‘†)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐿 ∈ 𝐷)    &   (πœ‘ β†’ 𝐸 = (π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©))    &   (πœ‘ β†’ 𝐹 = (π‘€β€˜βŸ¨β€œπΎπΏβ€βŸ©))    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐾 β‰  𝐿)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ Word 𝐢(𝐸 Β· 𝐹) = (𝑆 Ξ£g 𝑐))
 
Theoremcyc3genpm 31783* The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ (𝑄 ∈ 𝐴 ↔ βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀)))
 
Theoremcycpmgcl 31784 Cyclic permutations are permutations, similar to cycpmcl 31747, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ (0...𝑁)) β†’ 𝐢 βŠ† 𝐡)
 
Theoremcycpmconjslem1 31785 Lemma for cycpmconjs 31787. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ (β™―β€˜π‘Š) = 𝑃)    β‡’   (πœ‘ β†’ ((β—‘π‘Š ∘ (π‘€β€˜π‘Š)) ∘ π‘Š) = (( I β†Ύ (0..^𝑃)) cyclShift 1))
 
Theoremcycpmconjslem2 31786* Lemma for cycpmconjs 31787. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 𝑃 ∈ (0...𝑁))    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ž(π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
 
Theoremcycpmconjs 31787* All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 𝑃 ∈ (0...𝑁))    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    &   (πœ‘ β†’ 𝑇 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐡 𝑄 = ((𝑝 + 𝑇) βˆ’ 𝑝))
 
Theoremcyc3conja 31788* 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.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 5 ≀ 𝑁)    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    &   (πœ‘ β†’ 𝑇 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐴 𝑄 = ((𝑝 + 𝑇) βˆ’ 𝑝))
 
21.3.9.10  Signum in an ordered monoid
 
Syntaxcsgns 31789 Extend class notation to include the Signum function.
class sgns
 
Definitiondf-sgns 31790* Signum function for a structure. See also df-sgn 14906 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.)
sgns = (π‘Ÿ ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ if(π‘₯ = (0gβ€˜π‘Ÿ), 0, if((0gβ€˜π‘Ÿ)(ltβ€˜π‘Ÿ)π‘₯, 1, -1))))
 
Theoremsgnsv 31791* The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑆 = (π‘₯ ∈ 𝐡 ↦ if(π‘₯ = 0 , 0, if( 0 < π‘₯, 1, -1))))
 
Theoremsgnsval 31792 The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡) β†’ (π‘†β€˜π‘‹) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
 
Theoremsgnsf 31793 The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑆:𝐡⟢{-1, 0, 1})
 
21.3.9.11  The Archimedean property for generic ordered algebraic structures
 
Syntaxcinftm 31794 Class notation for the infinitesimal relation.
class β‹˜
 
Syntaxcarchi 31795 Class notation for the Archimedean property.
class Archi
 
Definitiondf-inftm 31796* Define the relation "π‘₯ is infinitesimal with respect to 𝑦 " for a structure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
β‹˜ = (𝑀 ∈ V ↦ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜π‘€) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) ∧ ((0gβ€˜π‘€)(ltβ€˜π‘€)π‘₯ ∧ βˆ€π‘› ∈ β„• (𝑛(.gβ€˜π‘€)π‘₯)(ltβ€˜π‘€)𝑦))})
 
Definitiondf-archi 31797 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi = {𝑀 ∣ (β‹˜β€˜π‘€) = βˆ…}
 
Theoreminftmrel 31798 The infinitesimal relation for a structure π‘Š. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    β‡’   (π‘Š ∈ 𝑉 β†’ (β‹˜β€˜π‘Š) βŠ† (𝐡 Γ— 𝐡))
 
Theoremisinftm 31799* Express π‘₯ is infinitesimal with respect to 𝑦 for a structure π‘Š. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    β‡’   ((π‘Š ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(β‹˜β€˜π‘Š)π‘Œ ↔ ( 0 < 𝑋 ∧ βˆ€π‘› ∈ β„• (𝑛 Β· 𝑋) < π‘Œ)))
 
Theoremisarchi 31800* Express the predicate "π‘Š is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (β‹˜β€˜π‘Š)    β‡’   (π‘Š ∈ 𝑉 β†’ (π‘Š ∈ Archi ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯ < 𝑦))
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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 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46966
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