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Type | Label | Description |
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Statement | ||
Theorem | omndmul2 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) & ⊢ · = (.g‘𝑀) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋)) | ||
Theorem | omndmul3 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) & ⊢ · = (.g‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑃 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ≤ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 0 ≤ 𝑋) ⇒ ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋)) | ||
Theorem | omndmul 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) & ⊢ · = (.g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑀 ∈ CMnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) ⇒ ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑁 · 𝑌)) | ||
Theorem | ogrpinv0le 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 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 )) | ||
Theorem | ogrpsub 32105 | In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ≤ = (le‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) ≤ (𝑌 − 𝑍)) | ||
Theorem | ogrpaddlt 32106 | In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍)) | ||
Theorem | ogrpaddltbi 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) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍))) | ||
Theorem | ogrpaddltrd 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) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 < 𝑌) ⇒ ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌)) | ||
Theorem | ogrpaddltrbid 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) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 < 𝑌 ↔ (𝑍 + 𝑋) < (𝑍 + 𝑌))) | ||
Theorem | ogrpsublt 32110 | In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 − 𝑍) < (𝑌 − 𝑍)) | ||
Theorem | ogrpinv0lt 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 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 )) | ||
Theorem | ogrpinvlt 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 = (invg‘𝐺) ⇒ ⊢ (((𝐺 ∈ oGrp ∧ (oppg‘𝐺) ∈ oGrp) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝐼‘𝑌) < (𝐼‘𝑋))) | ||
Theorem | gsumle 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) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑀 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 ∘r ≤ 𝐺) ⇒ ⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺)) | ||
Theorem | symgfcoeu 32114* | Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ 𝐺 = (Base‘(SymGrp‘𝐷)) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝐺 ∧ 𝑄 ∈ 𝐺) → ∃!𝑝 ∈ 𝐺 𝑄 = (𝑃 ∘ 𝑝)) | ||
Theorem | symgcom 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 ↾ 𝐸) = ( I ↾ 𝐸)) & ⊢ (𝜑 → (𝑌 ↾ 𝐹) = ( I ↾ 𝐹)) & ⊢ (𝜑 → (𝐸 ∩ 𝐹) = ∅) & ⊢ (𝜑 → (𝐸 ∪ 𝐹) = 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ∘ 𝑌) = (𝑌 ∘ 𝑋)) | ||
Theorem | symgcom2 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) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅) ⇒ ⊢ (𝜑 → (𝑋 ∘ 𝑌) = (𝑌 ∘ 𝑋)) | ||
Theorem | symgcntz 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) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 dom (𝑥 ∖ I )) ⇒ ⊢ (𝜑 → 𝐴 ⊆ (𝑍‘𝐴)) | ||
Theorem | odpmco 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 ∈ (𝐵 ∖ 𝐴) ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → (𝑋 ∘ 𝑌) ∈ 𝐴) | ||
Theorem | symgsubg 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) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 ∘ ◡𝑌)) | ||
Theorem | pmtrprfv2 32120 | In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ 𝑇 = (pmTrsp‘𝐷) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋) | ||
Theorem | pmtrcnel 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 = (𝐹‘𝐼) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I )) ⇒ ⊢ (𝜑 → dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∖ {𝐼})) | ||
Theorem | pmtrcnel2 32122 | Variation on pmtrcnel 32121. (Contributed by Thierry Arnoux, 16-Nov-2023.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑇 = (pmTrsp‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐽 = (𝐹‘𝐼) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I )) ⇒ ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) | ||
Theorem | pmtrcnelor 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 = (𝐹‘𝐼) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I )) & ⊢ 𝐸 = dom (𝐹 ∖ I ) & ⊢ 𝐴 = dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ⇒ ⊢ (𝜑 → (𝐴 = (𝐸 ∖ {𝐼, 𝐽}) ∨ 𝐴 = (𝐸 ∖ {𝐼}))) | ||
Theorem | pmtridf1o 32124 | Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴) | ||
Theorem | pmtridfv1 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑇‘𝑋) = 𝑌) | ||
Theorem | pmtridfv2 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) | ||
Theorem | psgnid 32127 | Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝑆 = (pmSgn‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) | ||
Theorem | psgndmfi 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) | ||
Theorem | pmtrto1cl 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) | ||
Theorem | psgnfzto1stlem 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) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≤ (𝐾 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝐾, (𝐾 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖 ≤ 𝐾, (𝑖 − 1), 𝑖))))) | ||
Theorem | fzto1stfv1 32131* | Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) ⇒ ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) | ||
Theorem | fzto1st1 32132* | Special case where the permutation defined in psgnfzto1st 32135 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) ⇒ ⊢ (𝐼 = 1 → 𝑃 = ( I ↾ 𝐷)) | ||
Theorem | fzto1st 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 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐼 ∈ 𝐷 → 𝑃 ∈ 𝐵) | ||
Theorem | fzto1stinvn 32134* | Value of the inverse of our permutation 𝑃 at 𝐼. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐼 ∈ 𝐷 → (◡𝑃‘𝐼) = 1) | ||
Theorem | psgnfzto1st 32135* | The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = (pmSgn‘𝐷) ⇒ ⊢ (𝐼 ∈ 𝐷 → (𝑆‘𝑃) = (-1↑(𝐼 + 1))) | ||
Syntax | ctocyc 32136 | Extend class notation with the permutation cycle builder. |
class toCyc | ||
Definition | df-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 𝑑 ∣ 𝑢:dom 𝑢–1-1→𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)))) | ||
Theorem | tocycval 32138* | Value of the cycle builder. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)))) | ||
Theorem | tocycfv 32139 | Function value of a permutation cycle built from a word. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) ⇒ ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) | ||
Theorem | tocycfvres1 32140 | A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) | ||
Theorem | tocycfvres2 32141 | A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊))) | ||
Theorem | cycpmfvlem 32142 | Lemma for cycpmfv1 32143 and cycpmfv2 32144. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊))) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) | ||
Theorem | cycpmfv1 32143 | Value of a cycle function for any element but the last. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 𝑁 ∈ (0..^((♯‘𝑊) − 1))) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘(𝑁 + 1))) | ||
Theorem | cycpmfv2 32144 | Value of a cycle function for the last element of the orbit. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 0 < (♯‘𝑊)) & ⊢ (𝜑 → 𝑁 = ((♯‘𝑊) − 1)) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘0)) | ||
Theorem | cycpmfv3 32145 | Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → ¬ 𝑋 ∈ ran 𝑊) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊)‘𝑋) = 𝑋) | ||
Theorem | cycpmcl 32146 | Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → (𝐶‘𝑊) ∈ (Base‘𝑆)) | ||
Theorem | tocycf 32147* | The permutation cycle builder as a function. (Contributed by Thierry Arnoux, 25-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝐶:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶𝐵) | ||
Theorem | tocyc01 32148 | Permutation cycles built from the empty set or a singleton are the identity. (Contributed by Thierry Arnoux, 21-Nov-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ (dom 𝐶 ∩ (◡♯ “ {0, 1}))) → (𝐶‘𝑊) = ( I ↾ 𝐷)) | ||
Theorem | cycpm2tr 32149 | A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑇 = (pmTrsp‘𝐷) ⇒ ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽”〉) = (𝑇‘{𝐼, 𝐽})) | ||
Theorem | cycpm2cl 32150 | Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽”〉) ∈ (Base‘𝑆)) | ||
Theorem | cyc2fv1 32151 | Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘𝐼) = 𝐽) | ||
Theorem | cyc2fv2 32152 | Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘𝐽) = 𝐼) | ||
Theorem | trsp2cyc 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) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘〈“𝑖𝑗”〉))) | ||
Theorem | cycpmco2f1 32154 | The word U used in cycpmco2 32163 is injective, so it can represent a cycle and form a cyclic permutation (𝑀‘𝑈). (Contributed by Thierry Arnoux, 4-Jan-2024.) |
⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) ⇒ ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷) | ||
Theorem | cycpmco2rn 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) ⇒ ⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼})) | ||
Theorem | cycpmco2lem1 32156 | Lemma for cycpmco2 32163. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) | ||
Theorem | cycpmco2lem2 32157 | Lemma for cycpmco2 32163. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) ⇒ ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼) | ||
Theorem | cycpmco2lem3 32158 | Lemma for cycpmco2 32163. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) ⇒ ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊)) | ||
Theorem | cycpmco2lem4 32159 | Lemma for cycpmco2 32163. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑈)‘𝐼)) | ||
Theorem | cycpmco2lem5 32160 | Lemma for cycpmco2 32163. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → (◡𝑈‘𝐾) = ((♯‘𝑈) − 1)) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
Theorem | cycpmco2lem6 32161 | Lemma for cycpmco2 32163. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) & ⊢ (𝜑 → (◡𝑈‘𝐾) ∈ (𝐸..^((♯‘𝑈) − 1))) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
Theorem | cycpmco2lem7 32162 | Lemma for cycpmco2 32163. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → 𝐾 ≠ 𝐽) & ⊢ (𝜑 → (◡𝑈‘𝐾) ∈ (0..^𝐸)) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
Theorem | cycpmco2 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘〈“𝐼𝐽”〉)) = (𝑀‘𝑈)) | ||
Theorem | cyc2fvx 32164 | Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘𝐾) = 𝐾) | ||
Theorem | cycpm3cl 32165 | Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽𝐾”〉) ∈ (Base‘𝑆)) | ||
Theorem | cycpm3cl2 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽𝐾”〉) ∈ (𝐶 “ (◡♯ “ {3}))) | ||
Theorem | cyc3fv1 32167 | Function value of a 3-cycle at the first point. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽𝐾”〉)‘𝐼) = 𝐽) | ||
Theorem | cyc3fv2 32168 | Function value of a 3-cycle at the second point. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽𝐾”〉)‘𝐽) = 𝐾) | ||
Theorem | cyc3fv3 32169 | Function value of a 3-cycle at the third point. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽𝐾”〉)‘𝐾) = 𝐼) | ||
Theorem | cyc3co2 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) & ⊢ · = (+g‘𝑆) ⇒ ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽𝐾”〉) = ((𝐶‘〈“𝐼𝐾”〉) · (𝐶‘〈“𝐼𝐽”〉))) | ||
Theorem | cycpmconjvlem 32171 | Lemma for cycpmconjv 32172. (Contributed by Thierry Arnoux, 9-Oct-2023.) |
⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐵 ⊆ 𝐷) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵)))) | ||
Theorem | cycpmconjv 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) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (𝑀‘(𝐺 ∘ 𝑊))) | ||
Theorem | cycpmrn 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 1 < (♯‘𝑊)) ⇒ ⊢ (𝜑 → ran 𝑊 = dom ((𝑀‘𝑊) ∖ I )) | ||
Theorem | tocyccntz 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ran 𝑥) & ⊢ (𝜑 → 𝐴 ⊆ dom 𝑀) ⇒ ⊢ (𝜑 → (𝑀 “ 𝐴) ⊆ (𝑍‘(𝑀 “ 𝐴))) | ||
Theorem | evpmval 32175 | Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) | ||
Theorem | cnmsgn0g 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 = (0g‘𝑈) | ||
Theorem | evpmsubg 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)) | ||
Theorem | evpmid 32178 | The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷)) | ||
Theorem | altgnsg 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)) | ||
Theorem | cyc3evpm 32180 | 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
⊢ 𝐶 = ((toCyc‘𝐷) “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴) | ||
Theorem | cyc3genpmlem 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) & ⊢ · = (+g‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐿 ∈ 𝐷) & ⊢ (𝜑 → 𝐸 = (𝑀‘〈“𝐼𝐽”〉)) & ⊢ (𝜑 → 𝐹 = (𝑀‘〈“𝐾𝐿”〉)) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐾 ≠ 𝐿) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ Word 𝐶(𝐸 · 𝐹) = (𝑆 Σg 𝑐)) | ||
Theorem | cyc3genpm 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 → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))) | ||
Theorem | cycpmgcl 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) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ (0...𝑁)) → 𝐶 ⊆ 𝐵) | ||
Theorem | cycpmconjslem1 32184 | Lemma for cycpmconjs 32186. (Contributed by Thierry Arnoux, 14-Oct-2023.) |
⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → (♯‘𝑊) = 𝑃) ⇒ ⊢ (𝜑 → ((◡𝑊 ∘ (𝑀‘𝑊)) ∘ 𝑊) = (( I ↾ (0..^𝑃)) cyclShift 1)) | ||
Theorem | cycpmconjslem2 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) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 𝑃 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto→𝐷 ∧ ((◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))) | ||
Theorem | cycpmconjs 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) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 𝑃 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) & ⊢ (𝜑 → 𝑇 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝)) | ||
Theorem | cyc3conja 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) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 5 ≤ 𝑁) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) & ⊢ (𝜑 → 𝑇 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 𝑄 = ((𝑝 + 𝑇) − 𝑝)) | ||
Syntax | csgns 32188 | Extend class notation to include the Signum function. |
class sgns | ||
Definition | df-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 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1)))) | ||
Theorem | sgnsv 32190* | The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ < = (lt‘𝑅) & ⊢ 𝑆 = (sgns‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | ||
Theorem | sgnsval 32191 | The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ < = (lt‘𝑅) & ⊢ 𝑆 = (sgns‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑆‘𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1))) | ||
Theorem | sgnsf 32192 | The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ < = (lt‘𝑅) & ⊢ 𝑆 = (sgns‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝑆:𝐵⟶{-1, 0, 1}) | ||
Syntax | cinftm 32193 | Class notation for the infinitesimal relation. |
class ⋘ | ||
Syntax | carchi 32194 | Class notation for the Archimedean property. |
class Archi | ||
Definition | df-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)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))}) | ||
Definition | df-archi 32196 | A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | ||
Theorem | inftmrel 32197 | The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) | ||
Theorem | isinftm 32198* | Express 𝑥 is infinitesimal with respect to 𝑦 for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ < = (lt‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))) | ||
Theorem | isarchi 32199* | Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ < = (⋘‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦)) | ||
Theorem | pnfinf 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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