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Theorem cycpmconjs 32750
Description: All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.)
Hypotheses
Ref Expression
cycpmconjs.c 𝐶 = (𝑀 “ (♯ “ {𝑃}))
cycpmconjs.s 𝑆 = (SymGrp‘𝐷)
cycpmconjs.n 𝑁 = (♯‘𝐷)
cycpmconjs.m 𝑀 = (toCyc‘𝐷)
cycpmconjs.b 𝐵 = (Base‘𝑆)
cycpmconjs.a + = (+g𝑆)
cycpmconjs.l = (-g𝑆)
cycpmconjs.p (𝜑𝑃 ∈ (0...𝑁))
cycpmconjs.d (𝜑𝐷 ∈ Fin)
cycpmconjs.q (𝜑𝑄𝐶)
cycpmconjs.t (𝜑𝑇𝐶)
Assertion
Ref Expression
cycpmconjs (𝜑 → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
Distinct variable groups:   + ,𝑝   ,𝑝   𝐵,𝑝   𝐷,𝑝   𝑀,𝑝   𝑁,𝑝   𝑃,𝑝   𝑄,𝑝   𝑇,𝑝   𝜑,𝑝
Allowed substitution hints:   𝐶(𝑝)   𝑆(𝑝)

Proof of Theorem cycpmconjs
Dummy variables 𝑞 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cycpmconjs.c . . 3 𝐶 = (𝑀 “ (♯ “ {𝑃}))
2 cycpmconjs.s . . 3 𝑆 = (SymGrp‘𝐷)
3 cycpmconjs.n . . 3 𝑁 = (♯‘𝐷)
4 cycpmconjs.m . . 3 𝑀 = (toCyc‘𝐷)
5 cycpmconjs.b . . 3 𝐵 = (Base‘𝑆)
6 cycpmconjs.a . . 3 + = (+g𝑆)
7 cycpmconjs.l . . 3 = (-g𝑆)
8 cycpmconjs.p . . 3 (𝜑𝑃 ∈ (0...𝑁))
9 cycpmconjs.d . . 3 (𝜑𝐷 ∈ Fin)
10 cycpmconjs.q . . 3 (𝜑𝑄𝐶)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10cycpmconjslem2 32749 . 2 (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
12 cycpmconjs.t . . . . . 6 (𝜑𝑇𝐶)
131, 2, 3, 4, 5, 6, 7, 8, 9, 12cycpmconjslem2 32749 . . . . 5 (𝜑 → ∃𝑡(𝑡:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
1413ad2antrr 723 . . . 4 (((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ∃𝑡(𝑡:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
159ad4antr 729 . . . . . . 7 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝐷 ∈ Fin)
16 simp-4r 781 . . . . . . . 8 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑞:(0..^𝑁)–1-1-onto𝐷)
17 f1ocnv 6845 . . . . . . . . 9 (𝑡:(0..^𝑁)–1-1-onto𝐷𝑡:𝐷1-1-onto→(0..^𝑁))
1817ad2antlr 724 . . . . . . . 8 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑡:𝐷1-1-onto→(0..^𝑁))
19 f1oco 6856 . . . . . . . 8 ((𝑞:(0..^𝑁)–1-1-onto𝐷𝑡:𝐷1-1-onto→(0..^𝑁)) → (𝑞𝑡):𝐷1-1-onto𝐷)
2016, 18, 19syl2anc 583 . . . . . . 7 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞𝑡):𝐷1-1-onto𝐷)
212, 5elsymgbas 19289 . . . . . . . 8 (𝐷 ∈ Fin → ((𝑞𝑡) ∈ 𝐵 ↔ (𝑞𝑡):𝐷1-1-onto𝐷))
2221biimpar 477 . . . . . . 7 ((𝐷 ∈ Fin ∧ (𝑞𝑡):𝐷1-1-onto𝐷) → (𝑞𝑡) ∈ 𝐵)
2315, 20, 22syl2anc 583 . . . . . 6 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞𝑡) ∈ 𝐵)
24 simpr 484 . . . . . . . . 9 ((((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑝 = (𝑞𝑡)) → 𝑝 = (𝑞𝑡))
2524oveq1d 7427 . . . . . . . 8 ((((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑝 = (𝑞𝑡)) → (𝑝 + 𝑇) = ((𝑞𝑡) + 𝑇))
2625, 24oveq12d 7430 . . . . . . 7 ((((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑝 = (𝑞𝑡)) → ((𝑝 + 𝑇) 𝑝) = (((𝑞𝑡) + 𝑇) (𝑞𝑡)))
2726eqeq2d 2742 . . . . . 6 ((((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑝 = (𝑞𝑡)) → (𝑄 = ((𝑝 + 𝑇) 𝑝) ↔ 𝑄 = (((𝑞𝑡) + 𝑇) (𝑞𝑡))))
28 simpllr 773 . . . . . . . . . 10 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))
29 simpr 484 . . . . . . . . . 10 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))
3028, 29eqtr4d 2774 . . . . . . . . 9 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞𝑄) ∘ 𝑞) = ((𝑡𝑇) ∘ 𝑡))
3130coeq1d 5861 . . . . . . . 8 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞) = (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞))
3231coeq2d 5862 . . . . . . 7 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞)))
33 coass 6264 . . . . . . . . 9 ((𝑞 ∘ (𝑞𝑄)) ∘ (𝑞𝑞)) = (𝑞 ∘ ((𝑞𝑄) ∘ (𝑞𝑞)))
34 coass 6264 . . . . . . . . . 10 ((𝑞𝑞) ∘ 𝑄) = (𝑞 ∘ (𝑞𝑄))
3534coeq1i 5859 . . . . . . . . 9 (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞)) = ((𝑞 ∘ (𝑞𝑄)) ∘ (𝑞𝑞))
36 coass 6264 . . . . . . . . . 10 (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞) = ((𝑞𝑄) ∘ (𝑞𝑞))
3736coeq2i 5860 . . . . . . . . 9 (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (𝑞 ∘ ((𝑞𝑄) ∘ (𝑞𝑞)))
3833, 35, 373eqtr4ri 2770 . . . . . . . 8 (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞))
39 f1ococnv2 6860 . . . . . . . . . . . . 13 (𝑞:(0..^𝑁)–1-1-onto𝐷 → (𝑞𝑞) = ( I ↾ 𝐷))
4016, 39syl 17 . . . . . . . . . . . 12 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞𝑞) = ( I ↾ 𝐷))
4140coeq1d 5861 . . . . . . . . . . 11 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞𝑞) ∘ 𝑄) = (( I ↾ 𝐷) ∘ 𝑄))
421, 2, 3, 4, 5cycpmgcl 32747 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) → 𝐶𝐵)
439, 8, 42syl2anc 583 . . . . . . . . . . . . . . 15 (𝜑𝐶𝐵)
4443, 10sseldd 3983 . . . . . . . . . . . . . 14 (𝜑𝑄𝐵)
452, 5elsymgbas 19289 . . . . . . . . . . . . . . 15 (𝐷 ∈ Fin → (𝑄𝐵𝑄:𝐷1-1-onto𝐷))
4645biimpa 476 . . . . . . . . . . . . . 14 ((𝐷 ∈ Fin ∧ 𝑄𝐵) → 𝑄:𝐷1-1-onto𝐷)
479, 44, 46syl2anc 583 . . . . . . . . . . . . 13 (𝜑𝑄:𝐷1-1-onto𝐷)
48 f1of 6833 . . . . . . . . . . . . 13 (𝑄:𝐷1-1-onto𝐷𝑄:𝐷𝐷)
49 fcoi2 6766 . . . . . . . . . . . . 13 (𝑄:𝐷𝐷 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
5047, 48, 493syl 18 . . . . . . . . . . . 12 (𝜑 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
5150ad4antr 729 . . . . . . . . . . 11 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
5241, 51eqtrd 2771 . . . . . . . . . 10 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞𝑞) ∘ 𝑄) = 𝑄)
5352, 40coeq12d 5864 . . . . . . . . 9 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞)) = (𝑄 ∘ ( I ↾ 𝐷)))
54 fcoi1 6765 . . . . . . . . . . 11 (𝑄:𝐷𝐷 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
5547, 48, 543syl 18 . . . . . . . . . 10 (𝜑 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
5655ad4antr 729 . . . . . . . . 9 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
5753, 56eqtrd 2771 . . . . . . . 8 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞)) = 𝑄)
5838, 57eqtrid 2783 . . . . . . 7 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = 𝑄)
59 coass 6264 . . . . . . . . 9 ((𝑞 ∘ (𝑡𝑇)) ∘ (𝑡𝑞)) = (𝑞 ∘ ((𝑡𝑇) ∘ (𝑡𝑞)))
60 coass 6264 . . . . . . . . . 10 ((𝑞𝑡) ∘ 𝑇) = (𝑞 ∘ (𝑡𝑇))
6160coeq1i 5859 . . . . . . . . 9 (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = ((𝑞 ∘ (𝑡𝑇)) ∘ (𝑡𝑞))
62 coass 6264 . . . . . . . . . 10 (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞) = ((𝑡𝑇) ∘ (𝑡𝑞))
6362coeq2i 5860 . . . . . . . . 9 (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞)) = (𝑞 ∘ ((𝑡𝑇) ∘ (𝑡𝑞)))
6459, 61, 633eqtr4i 2769 . . . . . . . 8 (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞))
6543, 12sseldd 3983 . . . . . . . . . . . 12 (𝜑𝑇𝐵)
6665ad4antr 729 . . . . . . . . . . 11 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑇𝐵)
672, 5, 6symgov 19299 . . . . . . . . . . 11 (((𝑞𝑡) ∈ 𝐵𝑇𝐵) → ((𝑞𝑡) + 𝑇) = ((𝑞𝑡) ∘ 𝑇))
6823, 66, 67syl2anc 583 . . . . . . . . . 10 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞𝑡) + 𝑇) = ((𝑞𝑡) ∘ 𝑇))
6968oveq1d 7427 . . . . . . . . 9 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞𝑡) + 𝑇) (𝑞𝑡)) = (((𝑞𝑡) ∘ 𝑇) (𝑞𝑡)))
702symggrp 19316 . . . . . . . . . . . . . 14 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
719, 70syl 17 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ Grp)
7271ad4antr 729 . . . . . . . . . . . 12 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑆 ∈ Grp)
735, 6grpcl 18869 . . . . . . . . . . . 12 ((𝑆 ∈ Grp ∧ (𝑞𝑡) ∈ 𝐵𝑇𝐵) → ((𝑞𝑡) + 𝑇) ∈ 𝐵)
7472, 23, 66, 73syl3anc 1370 . . . . . . . . . . 11 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞𝑡) + 𝑇) ∈ 𝐵)
7568, 74eqeltrrd 2833 . . . . . . . . . 10 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞𝑡) ∘ 𝑇) ∈ 𝐵)
762, 5, 7symgsubg 32683 . . . . . . . . . 10 ((((𝑞𝑡) ∘ 𝑇) ∈ 𝐵 ∧ (𝑞𝑡) ∈ 𝐵) → (((𝑞𝑡) ∘ 𝑇) (𝑞𝑡)) = (((𝑞𝑡) ∘ 𝑇) ∘ (𝑞𝑡)))
7775, 23, 76syl2anc 583 . . . . . . . . 9 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞𝑡) ∘ 𝑇) (𝑞𝑡)) = (((𝑞𝑡) ∘ 𝑇) ∘ (𝑞𝑡)))
78 cnvco 5885 . . . . . . . . . . . 12 (𝑞𝑡) = (𝑡𝑞)
79 f1orel 6836 . . . . . . . . . . . . . 14 (𝑡:(0..^𝑁)–1-1-onto𝐷 → Rel 𝑡)
80 dfrel2 6188 . . . . . . . . . . . . . 14 (Rel 𝑡𝑡 = 𝑡)
8179, 80sylib 217 . . . . . . . . . . . . 13 (𝑡:(0..^𝑁)–1-1-onto𝐷𝑡 = 𝑡)
8281coeq1d 5861 . . . . . . . . . . . 12 (𝑡:(0..^𝑁)–1-1-onto𝐷 → (𝑡𝑞) = (𝑡𝑞))
8378, 82eqtrid 2783 . . . . . . . . . . 11 (𝑡:(0..^𝑁)–1-1-onto𝐷(𝑞𝑡) = (𝑡𝑞))
8483coeq2d 5862 . . . . . . . . . 10 (𝑡:(0..^𝑁)–1-1-onto𝐷 → (((𝑞𝑡) ∘ 𝑇) ∘ (𝑞𝑡)) = (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)))
8584ad2antlr 724 . . . . . . . . 9 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞𝑡) ∘ 𝑇) ∘ (𝑞𝑡)) = (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)))
8669, 77, 853eqtrrd 2776 . . . . . . . 8 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = (((𝑞𝑡) + 𝑇) (𝑞𝑡)))
8764, 86eqtr3id 2785 . . . . . . 7 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞)) = (((𝑞𝑡) + 𝑇) (𝑞𝑡)))
8832, 58, 873eqtr3d 2779 . . . . . 6 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑄 = (((𝑞𝑡) + 𝑇) (𝑞𝑡)))
8923, 27, 88rspcedvd 3614 . . . . 5 (((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
9089anasss 466 . . . 4 ((((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ (𝑡:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))) → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
9114, 90exlimddv 1937 . . 3 (((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
9291anasss 466 . 2 ((𝜑 ∧ (𝑞:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))) → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
9311, 92exlimddv 1937 1 (𝜑 → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1780  wcel 2105  wrex 3069  cun 3946  wss 3948  {csn 4628   I cid 5573  ccnv 5675  cres 5678  cima 5679  ccom 5680  Rel wrel 5681  wf 6539  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7412  Fincfn 8945  0cc0 11116  1c1 11117  ...cfz 13491  ..^cfzo 13634  chash 14297   cyclShift ccsh 14745  Basecbs 17151  +gcplusg 17204  Grpcgrp 18861  -gcsg 18863  SymGrpcsymg 19282  toCycctocyc 32700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-oadd 8476  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-dju 9902  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-xnn0 12552  df-z 12566  df-uz 12830  df-rp 12982  df-fz 13492  df-fzo 13635  df-fl 13764  df-mod 13842  df-hash 14298  df-word 14472  df-concat 14528  df-substr 14598  df-pfx 14628  df-csh 14746  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-tset 17223  df-0g 17394  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-submnd 18712  df-efmnd 18792  df-grp 18864  df-minusg 18865  df-sbg 18866  df-symg 19283  df-tocyc 32701
This theorem is referenced by:  cyc3conja  32751
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