P. 329 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	symgcom 32801	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 32802	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 32803*	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 32804	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 32805	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 32806	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 32807	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 32808	Variation on pmtrcnel 32807. (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 32809	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})))
21.3.9.7  Transpositions
Theorem	pmtridf1o 32810	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 32811	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 32812	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)
21.3.9.8  Permutation Signs
Theorem	psgnid 32813	Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
⊢ 𝑆 = (pmSgn‘𝐷)    ⇒   ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)
Theorem	psgndmfi 32814	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 32815	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 32816*	Lemma for psgnfzto1st 32821. 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 32817*	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 32818*	Special case where the permutation defined in psgnfzto1st 32821 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 32819*	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 32820*	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 32821*	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)))
21.3.9.9  Permutation cycles
Syntax	ctocyc 32822	Extend class notation with the permutation cycle builder.
class toCyc
Definition	df-tocyc 32823*	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 32824*	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 32825	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 32826	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 32827	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 32828	Lemma for cycpmfv1 32829 and cycpmfv2 32830. (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 32829	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 32830	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 32831	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 32832	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 32833*	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 32834	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 32835	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 32836	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 32837	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 32838	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 32839*	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 32840	The word U used in cycpmco2 32849 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 32841	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 32842	Lemma for cycpmco2 32849. (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 32843	Lemma for cycpmco2 32849. (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 32844	Lemma for cycpmco2 32849. (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 32845	Lemma for cycpmco2 32849. (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 32846	Lemma for cycpmco2 32849. (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 32847	Lemma for cycpmco2 32849. (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 32848	Lemma for cycpmco2 32849. (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 32849	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 32850	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 32851	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 32852	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 32853	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 32854	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 32855	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 32856	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 32857	Lemma for cycpmconjv 32858. (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 32858	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 32859	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 32860*	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)))
21.3.9.10  The Alternating Group
Theorem	evpmval 32861	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 32862	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 32863	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 32864	The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
⊢ 𝑆 = (SymGrp‘𝐷)    ⇒   ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷))
Theorem	altgnsg 32865	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 32866	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 32867*	Lemma for cyc3genpm 32868. (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 32868*	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 32869	Cyclic permutations are permutations, similar to cycpmcl 32832, 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 32870	Lemma for cycpmconjs 32872. (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 32871*	Lemma for cycpmconjs 32872. (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 32872*	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 32873*	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))
21.3.9.11  Signum in an ordered monoid
Syntax	csgns 32874	Extend class notation to include the Signum function.
class sgns
Definition	df-sgns 32875*	Signum function for a structure. See also df-sgn 15061 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 32876*	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 32877	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 32878	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})
21.3.9.12  The Archimedean property for generic ordered algebraic structures
Syntax	cinftm 32879	Class notation for the infinitesimal relation.
class ⋘
Syntax	carchi 32880	Class notation for the Archimedean property.
class Archi
Definition	df-inftm 32881*	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 32882	A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
Theorem	inftmrel 32883	The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
Theorem	isinftm 32884*	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 32885*	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 32886	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)+∞)
Theorem	xrnarchi 32887	The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
⊢ ¬ ℝ*𝑠 ∈ Archi
Theorem	isarchi2 32888*	Alternative way to express the predicate "𝑊 is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    ⇒   ⊢ ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 ≤ (𝑛 · 𝑥))))
Theorem	submarchi 32889	A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi)
Theorem	isarchi3 32890*	This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    ⇒   ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))
Theorem	archirng 32891*	Property of Archimedean ordered groups, framing positive 𝑌 between multiples of 𝑋. (Contributed by Thierry Arnoux, 12-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑋)    &   ⊢ (𝜑 → 0 < 𝑌)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
Theorem	archirngz 32892*	Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑋)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
Theorem	archiexdiv 32893*	In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    ⇒   ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋))
Theorem	archiabllem1a 32894*	Lemma for archiabl 32901: In case an archimedean group 𝑊 admits a smallest positive element 𝑈, then any positive element 𝑋 of 𝑊 can be written as (𝑛 · 𝑈) with 𝑛 ∈ ℕ. Since the reciprocal holds for negative elements, 𝑊 is then isomorphic to ℤ. (Contributed by Thierry Arnoux, 12-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑋)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝑋 = (𝑛 · 𝑈))
Theorem	archiabllem1b 32895*	Lemma for archiabl 32901. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥)    ⇒   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
Theorem	archiabllem1 32896*	Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥)    ⇒   ⊢ (𝜑 → 𝑊 ∈ Abel)
Theorem	archiabllem2a 32897*	Lemma for archiabl 32901, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑋)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ (𝑐 + 𝑐) ≤ 𝑋))
Theorem	archiabllem2c 32898*	Lemma for archiabl 32901. (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ (𝑋 + 𝑌) < (𝑌 + 𝑋))
Theorem	archiabllem2b 32899*	Lemma for archiabl 32901. (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	archiabllem2 32900*	Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎))    ⇒   ⊢ (𝜑 → 𝑊 ∈ Abel)