P. 337 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33601-33700   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-tocyc 33601*	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 33602*	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 33603	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 33604	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 33605	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 33606	Lemma for cycpmfv1 33607 and cycpmfv2 33608. (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 33607	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 33608	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 33609	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 33610	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 33611*	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 33612	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 33613	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 33614	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 33615	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 33616	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 33617*	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 33618	The word U used in cycpmco2 33627 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 33619	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 33620	Lemma for cycpmco2 33627. (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 33621	Lemma for cycpmco2 33627. (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 33622	Lemma for cycpmco2 33627. (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 33623	Lemma for cycpmco2 33627. (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 33624	Lemma for cycpmco2 33627. (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 33625	Lemma for cycpmco2 33627. (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 33626	Lemma for cycpmco2 33627. (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 33627	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 33628	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 33629	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 33630	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 33631	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 33632	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 33633	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 33634	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 33635	Lemma for cycpmconjv 33636. (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 33636	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 33637	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 33638*	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)))
21.3.10.12  The Alternating Group
Theorem	evpmval 33639	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 33640	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 33641	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 33642	The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
⊢ 𝑆 = (SymGrp‘𝐷)    ⇒   ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷))
Theorem	altgnsg 33643	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 33644	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 33645*	Lemma for cyc3genpm 33646. (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 33646*	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 33647	Cyclic permutations are permutations, similar to cycpmcl 33610, 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 33648	Lemma for cycpmconjs 33650. (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 33649*	Lemma for cycpmconjs 33650. (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 33650*	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 33651*	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))
21.3.10.13  Signum in an ordered monoid
Syntax	csgns 33652	Extend class notation to include the Signum function.
class sgns
Definition	df-sgns 33653*	Signum function for a structure. See also df-sgn 15207 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 33654*	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 33655	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 33656	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})
21.3.10.14  Fixed points
Syntax	cfxp 33657	Extend class notation with the fixed points operation.
class FixPts
Definition	df-fxp 33658*	Define the set of fixed points left unchanged by a group action. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥})
Theorem	fxpval 33659*	Value of the set of fixed points. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
Theorem	fxpss 33660	The set of fixed points is a subset of the set acted upon. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐵FixPts𝐴) ⊆ 𝐵)
Theorem	fxpgaval 33661*	Value of the set of fixed points for a group action 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝑈 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    ⇒   ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
Theorem	isfxp 33662*	Property of being a fixed point. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝑈 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
Theorem	fxpgaeq 33663	A fixed point 𝑋 is invariant under group action 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝑈 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶FixPts𝐴))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋)
Theorem	conjga 33664*	Group conjugation induces a group action. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ − = (-g‘𝑀)    &   ⊢ ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))    ⇒   ⊢ (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵))
Theorem	cntrval2 33665*	Express the center 𝑍 of a group 𝑀 as the set of fixed points of the conjugation operation ⊕. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ − = (-g‘𝑀)    &   ⊢ ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))    &   ⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ (𝑀 ∈ Grp → 𝑍 = (𝐵FixPts ⊕ ))
Theorem	fxpsubm 33666*	Provided the group action 𝐴 induces monoid automorphisms, the set of fixed points of 𝐴 on a monoid 𝑊 is a submonoid, which could be called the fixed submonoid under 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = (Base‘𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 MndHom 𝑊))    ⇒   ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubMnd‘𝑊))
Theorem	fxpsubg 33667*	The fixed points of a group action 𝐴 on a group 𝑊 is a subgroup. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = (Base‘𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 GrpHom 𝑊))    ⇒   ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubGrp‘𝑊))
Theorem	fxpsubrg 33668*	The fixed points of a group action 𝐴 on a ring 𝑊 is a subgring. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = (Base‘𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 RingHom 𝑊))    ⇒   ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubRing‘𝑊))
Theorem	fxpsdrg 33669*	The fixed points of a group action 𝐴 on a division ring 𝑊 is a sub-division-ring. Since sub-division-rings of fields are subfields (see fldsdrgfld 21016), (𝐶FixPts𝐴) might be called the fixed subfield under 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = (Base‘𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 RingHom 𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ DivRing)    ⇒   ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubDRing‘𝑊))
21.3.10.15  The Archimedean property for generic ordered algebraic structures
Syntax	cinftm 33670	Class notation for the infinitesimal relation.
class ⋘
Syntax	carchi 33671	Class notation for the Archimedean property.
class Archi
Definition	df-inftm 33672*	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 33673	A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
Theorem	inftmrel 33674	The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
Theorem	isinftm 33675*	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 33676*	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 33677	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 33678	The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
⊢ ¬ ℝ*𝑠 ∈ Archi
Theorem	isarchi2 33679*	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 33680	A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi)
Theorem	isarchi3 33681*	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 33682*	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 33683*	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 33684*	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 33685*	Lemma for archiabl 33692: 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 33686*	Lemma for archiabl 33692. (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 33687*	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 33688*	Lemma for archiabl 33692, 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 33689*	Lemma for archiabl 33692. (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 33690*	Lemma for archiabl 33692. (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 33691*	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)
Theorem	archiabl 33692	Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
Theorem	isarchiofld 33693*	Axiom of Archimedes : a characterization of the Archimedean property for ordered fields. (Contributed by Thierry Arnoux, 9-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐻 = (ℤRHom‘𝑊)    &   ⊢ < = (lt‘𝑊)    ⇒   ⊢ (𝑊 ∈ oField → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∃𝑛 ∈ ℕ 𝑥 < (𝐻‘𝑛)))
21.3.10.16  Semiring left modules
Syntax	cslmd 33694	Extend class notation with class of all semimodules.
class SLMod
Definition	df-slmd 33695*	Define the class of all (left) modules over semirings, i.e. semimodules, which are generalizations of left modules. A semimodule is a commutative monoid (=vectors) together with a semiring (=scalars) and a left scalar product connecting them. (0[,]+∞) for example is not a full fledged left module, but is a semimodule. Definition of [Golan] p. 149. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ SLMod = {𝑔 ∈ CMnd ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][( ·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))))}
Theorem	isslmd 33696*	The predicate "is a semimodule". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ × = (.r‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 𝑂 = (0g‘𝐹)    ⇒   ⊢ (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤 ∧ (𝑂 · 𝑤) = 0 ))))
Theorem	slmdlema 33697	Lemma for properties of a semimodule. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ × = (.r‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 𝑂 = (0g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑅 · 𝑌) ∈ 𝑉 ∧ (𝑅 · (𝑌 + 𝑋)) = ((𝑅 · 𝑌) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑌) = ((𝑄 · 𝑌) + (𝑅 · 𝑌))) ∧ (((𝑄 × 𝑅) · 𝑌) = (𝑄 · (𝑅 · 𝑌)) ∧ ( 1 · 𝑌) = 𝑌 ∧ (𝑂 · 𝑌) = 0 )))
Theorem	lmodslmd 33698	Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ (𝑊 ∈ LMod → 𝑊 ∈ SLMod)
Theorem	slmdcmn 33699	A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
Theorem	slmdmnd 33700	A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)