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Theorem cycpmconjs 33151
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 33150 . 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 33150 . . . . 5 (𝜑 → ∃𝑡(𝑡:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑡𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
1413ad2antrr 725 . . . 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 731 . . . . . . 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 783 . . . . . . . 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 6876 . . . . . . . . 9 (𝑡:(0..^𝑁)–1-1-onto𝐷𝑡:𝐷1-1-onto→(0..^𝑁))
1817ad2antlr 726 . . . . . . . 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 6887 . . . . . . . 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 19417 . . . . . . . 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 7465 . . . . . . . 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 7468 . . . . . . 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 2751 . . . . . 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 775 . . . . . . . . . 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 2783 . . . . . . . . 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 5886 . . . . . . . 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 5887 . . . . . . 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 6298 . . . . . . . . 9 ((𝑞 ∘ (𝑞𝑄)) ∘ (𝑞𝑞)) = (𝑞 ∘ ((𝑞𝑄) ∘ (𝑞𝑞)))
34 coass 6298 . . . . . . . . . 10 ((𝑞𝑞) ∘ 𝑄) = (𝑞 ∘ (𝑞𝑄))
3534coeq1i 5884 . . . . . . . . 9 (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞)) = ((𝑞 ∘ (𝑞𝑄)) ∘ (𝑞𝑞))
36 coass 6298 . . . . . . . . . 10 (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞) = ((𝑞𝑄) ∘ (𝑞𝑞))
3736coeq2i 5885 . . . . . . . . 9 (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (𝑞 ∘ ((𝑞𝑄) ∘ (𝑞𝑞)))
3833, 35, 373eqtr4ri 2779 . . . . . . . 8 (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞))
39 f1ococnv2 6891 . . . . . . . . . . . . 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 5886 . . . . . . . . . . 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 33148 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) → 𝐶𝐵)
439, 8, 42syl2anc 583 . . . . . . . . . . . . . . 15 (𝜑𝐶𝐵)
4443, 10sseldd 4009 . . . . . . . . . . . . . 14 (𝜑𝑄𝐵)
452, 5elsymgbas 19417 . . . . . . . . . . . . . . 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 6864 . . . . . . . . . . . . 13 (𝑄:𝐷1-1-onto𝐷𝑄:𝐷𝐷)
49 fcoi2 6798 . . . . . . . . . . . . 13 (𝑄:𝐷𝐷 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
5047, 48, 493syl 18 . . . . . . . . . . . 12 (𝜑 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
5150ad4antr 731 . . . . . . . . . . 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 2780 . . . . . . . . . 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 5889 . . . . . . . . 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 6797 . . . . . . . . . . 11 (𝑄:𝐷𝐷 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
5547, 48, 543syl 18 . . . . . . . . . 10 (𝜑 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
5655ad4antr 731 . . . . . . . . 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 2780 . . . . . . . 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 2792 . . . . . . 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 6298 . . . . . . . . 9 ((𝑞 ∘ (𝑡𝑇)) ∘ (𝑡𝑞)) = (𝑞 ∘ ((𝑡𝑇) ∘ (𝑡𝑞)))
60 coass 6298 . . . . . . . . . 10 ((𝑞𝑡) ∘ 𝑇) = (𝑞 ∘ (𝑡𝑇))
6160coeq1i 5884 . . . . . . . . 9 (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = ((𝑞 ∘ (𝑡𝑇)) ∘ (𝑡𝑞))
62 coass 6298 . . . . . . . . . 10 (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞) = ((𝑡𝑇) ∘ (𝑡𝑞))
6362coeq2i 5885 . . . . . . . . 9 (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞)) = (𝑞 ∘ ((𝑡𝑇) ∘ (𝑡𝑞)))
6459, 61, 633eqtr4i 2778 . . . . . . . 8 (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞))
6543, 12sseldd 4009 . . . . . . . . . . . 12 (𝜑𝑇𝐵)
6665ad4antr 731 . . . . . . . . . . 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 19427 . . . . . . . . . . 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 7465 . . . . . . . . 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 19444 . . . . . . . . . . . . . 14 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
719, 70syl 17 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ Grp)
7271ad4antr 731 . . . . . . . . . . . 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 18983 . . . . . . . . . . . 12 ((𝑆 ∈ Grp ∧ (𝑞𝑡) ∈ 𝐵𝑇𝐵) → ((𝑞𝑡) + 𝑇) ∈ 𝐵)
7472, 23, 66, 73syl3anc 1371 . . . . . . . . . . 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 2845 . . . . . . . . . 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 33082 . . . . . . . . . 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 5910 . . . . . . . . . . . 12 (𝑞𝑡) = (𝑡𝑞)
79 f1orel 6867 . . . . . . . . . . . . . 14 (𝑡:(0..^𝑁)–1-1-onto𝐷 → Rel 𝑡)
80 dfrel2 6222 . . . . . . . . . . . . . 14 (Rel 𝑡𝑡 = 𝑡)
8179, 80sylib 218 . . . . . . . . . . . . 13 (𝑡:(0..^𝑁)–1-1-onto𝐷𝑡 = 𝑡)
8281coeq1d 5886 . . . . . . . . . . . 12 (𝑡:(0..^𝑁)–1-1-onto𝐷 → (𝑡𝑞) = (𝑡𝑞))
8378, 82eqtrid 2792 . . . . . . . . . . 11 (𝑡:(0..^𝑁)–1-1-onto𝐷(𝑞𝑡) = (𝑡𝑞))
8483coeq2d 5887 . . . . . . . . . 10 (𝑡:(0..^𝑁)–1-1-onto𝐷 → (((𝑞𝑡) ∘ 𝑇) ∘ (𝑞𝑡)) = (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)))
8584ad2antlr 726 . . . . . . . . 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 2785 . . . . . . . 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 2794 . . . . . . 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 2788 . . . . . 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 3637 . . . . 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 1934 . . 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 1934 1 (𝜑 → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1777  wcel 2108  wrex 3076  cun 3974  wss 3976  {csn 4648   I cid 5592  ccnv 5699  cres 5702  cima 5703  ccom 5704  Rel wrel 5705  wf 6571  1-1-ontowf1o 6574  cfv 6575  (class class class)co 7450  Fincfn 9005  0cc0 11186  1c1 11187  ...cfz 13569  ..^cfzo 13713  chash 14381   cyclShift ccsh 14838  Basecbs 17260  +gcplusg 17313  Grpcgrp 18975  -gcsg 18977  SymGrpcsymg 19412  toCycctocyc 33101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-cnex 11242  ax-resscn 11243  ax-1cn 11244  ax-icn 11245  ax-addcl 11246  ax-addrcl 11247  ax-mulcl 11248  ax-mulrcl 11249  ax-mulcom 11250  ax-addass 11251  ax-mulass 11252  ax-distr 11253  ax-i2m1 11254  ax-1ne0 11255  ax-1rid 11256  ax-rnegex 11257  ax-rrecex 11258  ax-cnre 11259  ax-pre-lttri 11260  ax-pre-lttrn 11261  ax-pre-ltadd 11262  ax-pre-mulgt0 11263  ax-pre-sup 11264
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-1st 8032  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-1o 8524  df-2o 8525  df-oadd 8528  df-er 8765  df-map 8888  df-en 9006  df-dom 9007  df-sdom 9008  df-fin 9009  df-sup 9513  df-inf 9514  df-dju 9972  df-card 10010  df-pnf 11328  df-mnf 11329  df-xr 11330  df-ltxr 11331  df-le 11332  df-sub 11524  df-neg 11525  df-div 11950  df-nn 12296  df-2 12358  df-3 12359  df-4 12360  df-5 12361  df-6 12362  df-7 12363  df-8 12364  df-9 12365  df-n0 12556  df-xnn0 12628  df-z 12642  df-uz 12906  df-rp 13060  df-fz 13570  df-fzo 13714  df-fl 13845  df-mod 13923  df-hash 14382  df-word 14565  df-concat 14621  df-substr 14691  df-pfx 14721  df-csh 14839  df-struct 17196  df-sets 17213  df-slot 17231  df-ndx 17243  df-base 17261  df-ress 17290  df-plusg 17326  df-tset 17332  df-0g 17503  df-mgm 18680  df-sgrp 18759  df-mnd 18775  df-submnd 18821  df-efmnd 18906  df-grp 18978  df-minusg 18979  df-sbg 18980  df-symg 19413  df-tocyc 33102
This theorem is referenced by:  cyc3conja  33152
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