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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tocycfv 30801 | 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 30802 | 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 30803 | 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 30804 | Lemma for cycpmfv1 30805 and cycpmfv2 30806. (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 30805 | 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 30806 | 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 30807 | 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 30808 | 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 30809* | 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 30810 | 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 30811 | 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 30812 | 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 30813 | 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 30814 | 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 30815* | 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 30816 | The word U used in cycpmco2 30825 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 30817 | 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 30818 | Lemma for cycpmco2 30825. (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 30819 | Lemma for cycpmco2 30825. (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 30820 | Lemma for cycpmco2 30825. (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 30821 | Lemma for cycpmco2 30825. (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 30822 | Lemma for cycpmco2 30825. (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 30823 | Lemma for cycpmco2 30825. (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 30824 | Lemma for cycpmco2 30825. (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 30825 | 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 30826 | 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 30827 | 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 30828 | 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 30829 | 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 30830 | 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 30831 | 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 30832 | 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 30833 | Lemma for cycpmconjv 30834. (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 30834 | 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 30835 | 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 30836* | 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 30837 | 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 30838 | 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 30839 | 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 30840 | The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷)) | ||
Theorem | altgnsg 30841 | 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 30842 | 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 30843* | Lemma for cyc3genpm 30844. (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 30844* | 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 30845 | Cyclic permutations are permutations, similar to cycpmcl 30808, 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 30846 | Lemma for cycpmconjs 30848. (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 30847* | Lemma for cycpmconjs 30848. (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 30848* | 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 30849* | 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 30850 | Extend class notation to include the Signum function. |
class sgns | ||
Definition | df-sgns 30851* | Signum function for a structure. See also df-sgn 14438 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 30852* | 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 30853 | 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 30854 | 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 30855 | Class notation for the infinitesimal relation. |
class ⋘ | ||
Syntax | carchi 30856 | Class notation for the Archimedean property. |
class Archi | ||
Definition | df-inftm 30857* | 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 30858 | A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | ||
Theorem | inftmrel 30859 | The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) | ||
Theorem | isinftm 30860* | 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 30861* | 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 30862 | 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 30863 | The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.) |
⊢ ¬ ℝ*𝑠 ∈ Archi | ||
Theorem | isarchi2 30864* | 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 30865 | A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.) |
⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi) | ||
Theorem | isarchi3 30866* | 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 30867* | 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 30868* | 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 30869* | 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 30870* | Lemma for archiabl 30877: 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 30871* | Lemma for archiabl 30877. (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 30872* | 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 30873* | Lemma for archiabl 30877, 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 30874* | Lemma for archiabl 30877. (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 30875* | Lemma for archiabl 30877. (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 30876* | 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 30877 | Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.) |
⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel) | ||
Syntax | cslmd 30878 | Extend class notation with class of all semimodules. |
class SLMod | ||
Definition | df-slmd 30879* | 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 30880* | 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 30881 | 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 30882 | Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ (𝑊 ∈ LMod → 𝑊 ∈ SLMod) | ||
Theorem | slmdcmn 30883 | A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd) | ||
Theorem | slmdmnd 30884 | A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | ||
Theorem | slmdsrg 30885 | The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) | ||
Theorem | slmdbn0 30886 | The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) | ||
Theorem | slmdacl 30887 | Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ + = (+g‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) | ||
Theorem | slmdmcl 30888 | Closure of ring multiplication for a semimodule. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = (.r‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 · 𝑌) ∈ 𝐾) | ||
Theorem | slmdsn0 30889 | The set of scalars in a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) | ||
Theorem | slmdvacl 30890 | Closure of vector addition for a semiring left module. (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) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) | ||
Theorem | slmdass 30891 | Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) | ||
Theorem | slmdvscl 30892 | Closure of scalar product for a semiring left module. (hvmulcl 28796 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉) | ||
Theorem | slmdvsdi 30893 | Distributive law for scalar product. (ax-hvdistr1 28791 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) | ||
Theorem | slmdvsdir 30894 | Distributive law for scalar product. (ax-hvdistr1 28791 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) | ||
Theorem | slmdvsass 30895 | Associative law for scalar product. (ax-hvmulass 28790 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ × = (.r‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) | ||
Theorem | slmd0cl 30896 | The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾) | ||
Theorem | slmd1cl 30897 | The ring unit in a semiring left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod → 1 ∈ 𝐾) | ||
Theorem | slmdvs1 30898 | Scalar product with ring unit. (ax-hvmulid 28789 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋) | ||
Theorem | slmd0vcl 30899 | The zero vector is a vector. (ax-hv0cl 28786 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝑉) | ||
Theorem | slmd0vlid 30900 | Left identity law for the zero vector. (hvaddid2 28806 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) |
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