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Theorem cycpmconjs 30962
 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 30961 . 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 30961 . . . . 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 6619 . . . . . . . . 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 6629 . . . . . . . 8 ((𝑞:(0..^𝑁)–1-1-onto𝐷𝑡:𝐷1-1-onto→(0..^𝑁)) → (𝑞𝑡):𝐷1-1-onto𝐷)
2016, 18, 19syl2anc 587 . . . . . . 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 18583 . . . . . . . 8 (𝐷 ∈ Fin → ((𝑞𝑡) ∈ 𝐵 ↔ (𝑞𝑡):𝐷1-1-onto𝐷))
2221biimpar 481 . . . . . . 7 ((𝐷 ∈ Fin ∧ (𝑞𝑡):𝐷1-1-onto𝐷) → (𝑞𝑡) ∈ 𝐵)
2315, 20, 22syl2anc 587 . . . . . 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 488 . . . . . . . . 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 7171 . . . . . . . 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 7174 . . . . . . 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 2769 . . . . . 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 488 . . . . . . . . . 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 2796 . . . . . . . . 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 5707 . . . . . . . 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 5708 . . . . . . 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 6100 . . . . . . . . 9 ((𝑞 ∘ (𝑞𝑄)) ∘ (𝑞𝑞)) = (𝑞 ∘ ((𝑞𝑄) ∘ (𝑞𝑞)))
34 coass 6100 . . . . . . . . . 10 ((𝑞𝑞) ∘ 𝑄) = (𝑞 ∘ (𝑞𝑄))
3534coeq1i 5705 . . . . . . . . 9 (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞)) = ((𝑞 ∘ (𝑞𝑄)) ∘ (𝑞𝑞))
36 coass 6100 . . . . . . . . . 10 (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞) = ((𝑞𝑄) ∘ (𝑞𝑞))
3736coeq2i 5706 . . . . . . . . 9 (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (𝑞 ∘ ((𝑞𝑄) ∘ (𝑞𝑞)))
3833, 35, 373eqtr4ri 2792 . . . . . . . 8 (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞))
39 f1ococnv2 6633 . . . . . . . . . . . . 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 5707 . . . . . . . . . . 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 30959 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) → 𝐶𝐵)
439, 8, 42syl2anc 587 . . . . . . . . . . . . . . 15 (𝜑𝐶𝐵)
4443, 10sseldd 3895 . . . . . . . . . . . . . 14 (𝜑𝑄𝐵)
452, 5elsymgbas 18583 . . . . . . . . . . . . . . 15 (𝐷 ∈ Fin → (𝑄𝐵𝑄:𝐷1-1-onto𝐷))
4645biimpa 480 . . . . . . . . . . . . . 14 ((𝐷 ∈ Fin ∧ 𝑄𝐵) → 𝑄:𝐷1-1-onto𝐷)
479, 44, 46syl2anc 587 . . . . . . . . . . . . 13 (𝜑𝑄:𝐷1-1-onto𝐷)
48 f1of 6607 . . . . . . . . . . . . 13 (𝑄:𝐷1-1-onto𝐷𝑄:𝐷𝐷)
49 fcoi2 6543 . . . . . . . . . . . . 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 2793 . . . . . . . . . 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 5710 . . . . . . . . 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 6542 . . . . . . . . . . 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 2793 . . . . . . . 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, 57syl5eq 2805 . . . . . . 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 6100 . . . . . . . . 9 ((𝑞 ∘ (𝑡𝑇)) ∘ (𝑡𝑞)) = (𝑞 ∘ ((𝑡𝑇) ∘ (𝑡𝑞)))
60 coass 6100 . . . . . . . . . 10 ((𝑞𝑡) ∘ 𝑇) = (𝑞 ∘ (𝑡𝑇))
6160coeq1i 5705 . . . . . . . . 9 (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = ((𝑞 ∘ (𝑡𝑇)) ∘ (𝑡𝑞))
62 coass 6100 . . . . . . . . . 10 (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞) = ((𝑡𝑇) ∘ (𝑡𝑞))
6362coeq2i 5706 . . . . . . . . 9 (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞)) = (𝑞 ∘ ((𝑡𝑇) ∘ (𝑡𝑞)))
6459, 61, 633eqtr4i 2791 . . . . . . . 8 (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞))
6543, 12sseldd 3895 . . . . . . . . . . . 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 18593 . . . . . . . . . . 11 (((𝑞𝑡) ∈ 𝐵𝑇𝐵) → ((𝑞𝑡) + 𝑇) = ((𝑞𝑡) ∘ 𝑇))
6823, 66, 67syl2anc 587 . . . . . . . . . 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 7171 . . . . . . . . 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 18609 . . . . . . . . . . . . . 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 18191 . . . . . . . . . . . 12 ((𝑆 ∈ Grp ∧ (𝑞𝑡) ∈ 𝐵𝑇𝐵) → ((𝑞𝑡) + 𝑇) ∈ 𝐵)
7472, 23, 66, 73syl3anc 1368 . . . . . . . . . . 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 2853 . . . . . . . . . 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 30895 . . . . . . . . . 10 ((((𝑞𝑡) ∘ 𝑇) ∈ 𝐵 ∧ (𝑞𝑡) ∈ 𝐵) → (((𝑞𝑡) ∘ 𝑇) (𝑞𝑡)) = (((𝑞𝑡) ∘ 𝑇) ∘ (𝑞𝑡)))
7775, 23, 76syl2anc 587 . . . . . . . . 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 5731 . . . . . . . . . . . 12 (𝑞𝑡) = (𝑡𝑞)
79 f1orel 6610 . . . . . . . . . . . . . 14 (𝑡:(0..^𝑁)–1-1-onto𝐷 → Rel 𝑡)
80 dfrel2 6023 . . . . . . . . . . . . . 14 (Rel 𝑡𝑡 = 𝑡)
8179, 80sylib 221 . . . . . . . . . . . . 13 (𝑡:(0..^𝑁)–1-1-onto𝐷𝑡 = 𝑡)
8281coeq1d 5707 . . . . . . . . . . . 12 (𝑡:(0..^𝑁)–1-1-onto𝐷 → (𝑡𝑞) = (𝑡𝑞))
8378, 82syl5eq 2805 . . . . . . . . . . 11 (𝑡:(0..^𝑁)–1-1-onto𝐷(𝑞𝑡) = (𝑡𝑞))
8483coeq2d 5708 . . . . . . . . . 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 2798 . . . . . . . 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 2807 . . . . . . 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 2801 . . . . . 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 3546 . . . . 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 470 . . . 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 1936 . . 3 (((𝜑𝑞:(0..^𝑁)–1-1-onto𝐷) ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
9291anasss 470 . 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 1936 1 (𝜑 → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071   ∪ cun 3858   ⊆ wss 3860  {csn 4525   I cid 5433  ◡ccnv 5527   ↾ cres 5530   “ cima 5531   ∘ ccom 5532  Rel wrel 5533  ⟶wf 6336  –1-1-onto→wf1o 6339  ‘cfv 6340  (class class class)co 7156  Fincfn 8540  0cc0 10588  1c1 10589  ...cfz 12952  ..^cfzo 13095  ♯chash 13753   cyclShift ccsh 14210  Basecbs 16555  +gcplusg 16637  Grpcgrp 18183  -gcsg 18185  SymGrpcsymg 18576  toCycctocyc 30912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665  ax-pre-sup 10666 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-oadd 8122  df-er 8305  df-map 8424  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-sup 8952  df-inf 8953  df-dju 9376  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349  df-nn 11688  df-2 11750  df-3 11751  df-4 11752  df-5 11753  df-6 11754  df-7 11755  df-8 11756  df-9 11757  df-n0 11948  df-xnn0 12020  df-z 12034  df-uz 12296  df-rp 12444  df-fz 12953  df-fzo 13096  df-fl 13224  df-mod 13300  df-hash 13754  df-word 13927  df-concat 13983  df-substr 14063  df-pfx 14093  df-csh 14211  df-struct 16557  df-ndx 16558  df-slot 16559  df-base 16561  df-sets 16562  df-ress 16563  df-plusg 16650  df-tset 16656  df-0g 16787  df-mgm 17932  df-sgrp 17981  df-mnd 17992  df-submnd 18037  df-efmnd 18114  df-grp 18186  df-minusg 18187  df-sbg 18188  df-symg 18577  df-tocyc 30913 This theorem is referenced by:  cyc3conja  30963
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