Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cycpmconjs Structured version   Visualization version   GIF version

Theorem cycpmconjs 33141
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 33140 . 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 33140 . . . . 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 6873 . . . . . . . . 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 6884 . . . . . . . 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 19410 . . . . . . . 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 7460 . . . . . . . 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 7463 . . . . . . 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 2745 . . . . . 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 2777 . . . . . . . . 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 5885 . . . . . . . 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 5886 . . . . . . 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 6295 . . . . . . . . 9 ((𝑞 ∘ (𝑞𝑄)) ∘ (𝑞𝑞)) = (𝑞 ∘ ((𝑞𝑄) ∘ (𝑞𝑞)))
34 coass 6295 . . . . . . . . . 10 ((𝑞𝑞) ∘ 𝑄) = (𝑞 ∘ (𝑞𝑄))
3534coeq1i 5883 . . . . . . . . 9 (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞)) = ((𝑞 ∘ (𝑞𝑄)) ∘ (𝑞𝑞))
36 coass 6295 . . . . . . . . . 10 (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞) = ((𝑞𝑄) ∘ (𝑞𝑞))
3736coeq2i 5884 . . . . . . . . 9 (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (𝑞 ∘ ((𝑞𝑄) ∘ (𝑞𝑞)))
3833, 35, 373eqtr4ri 2773 . . . . . . . 8 (𝑞 ∘ (((𝑞𝑄) ∘ 𝑞) ∘ 𝑞)) = (((𝑞𝑞) ∘ 𝑄) ∘ (𝑞𝑞))
39 f1ococnv2 6888 . . . . . . . . . . . . 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 5885 . . . . . . . . . . 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 33138 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) → 𝐶𝐵)
439, 8, 42syl2anc 583 . . . . . . . . . . . . . . 15 (𝜑𝐶𝐵)
4443, 10sseldd 4003 . . . . . . . . . . . . . 14 (𝜑𝑄𝐵)
452, 5elsymgbas 19410 . . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . 13 (𝑄:𝐷1-1-onto𝐷𝑄:𝐷𝐷)
49 fcoi2 6795 . . . . . . . . . . . . 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 2774 . . . . . . . . . 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 5888 . . . . . . . . 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 6794 . . . . . . . . . . 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 2774 . . . . . . . 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 2786 . . . . . . 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 6295 . . . . . . . . 9 ((𝑞 ∘ (𝑡𝑇)) ∘ (𝑡𝑞)) = (𝑞 ∘ ((𝑡𝑇) ∘ (𝑡𝑞)))
60 coass 6295 . . . . . . . . . 10 ((𝑞𝑡) ∘ 𝑇) = (𝑞 ∘ (𝑡𝑇))
6160coeq1i 5883 . . . . . . . . 9 (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = ((𝑞 ∘ (𝑡𝑇)) ∘ (𝑡𝑞))
62 coass 6295 . . . . . . . . . 10 (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞) = ((𝑡𝑇) ∘ (𝑡𝑞))
6362coeq2i 5884 . . . . . . . . 9 (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞)) = (𝑞 ∘ ((𝑡𝑇) ∘ (𝑡𝑞)))
6459, 61, 633eqtr4i 2772 . . . . . . . 8 (((𝑞𝑡) ∘ 𝑇) ∘ (𝑡𝑞)) = (𝑞 ∘ (((𝑡𝑇) ∘ 𝑡) ∘ 𝑞))
6543, 12sseldd 4003 . . . . . . . . . . . 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 19420 . . . . . . . . . . 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 7460 . . . . . . . . 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 19437 . . . . . . . . . . . . . 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 18976 . . . . . . . . . . . 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 2839 . . . . . . . . . 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 33072 . . . . . . . . . 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 5909 . . . . . . . . . . . 12 (𝑞𝑡) = (𝑡𝑞)
79 f1orel 6864 . . . . . . . . . . . . . 14 (𝑡:(0..^𝑁)–1-1-onto𝐷 → Rel 𝑡)
80 dfrel2 6219 . . . . . . . . . . . . . 14 (Rel 𝑡𝑡 = 𝑡)
8179, 80sylib 218 . . . . . . . . . . . . 13 (𝑡:(0..^𝑁)–1-1-onto𝐷𝑡 = 𝑡)
8281coeq1d 5885 . . . . . . . . . . . 12 (𝑡:(0..^𝑁)–1-1-onto𝐷 → (𝑡𝑞) = (𝑡𝑞))
8378, 82eqtrid 2786 . . . . . . . . . . 11 (𝑡:(0..^𝑁)–1-1-onto𝐷(𝑞𝑡) = (𝑡𝑞))
8483coeq2d 5886 . . . . . . . . . 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 2779 . . . . . . . 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 2788 . . . . . . 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 2782 . . . . . 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 3633 . . . . 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 2103  wrex 3072  cun 3968  wss 3970  {csn 4648   I cid 5596  ccnv 5698  cres 5701  cima 5702  ccom 5703  Rel wrel 5704  wf 6568  1-1-ontowf1o 6571  cfv 6572  (class class class)co 7445  Fincfn 8999  0cc0 11180  1c1 11181  ...cfz 13563  ..^cfzo 13707  chash 14375   cyclShift ccsh 14832  Basecbs 17253  +gcplusg 17306  Grpcgrp 18968  -gcsg 18970  SymGrpcsymg 19405  toCycctocyc 33091
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257  ax-pre-sup 11258
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-oadd 8522  df-er 8759  df-map 8882  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-sup 9507  df-inf 9508  df-dju 9966  df-card 10004  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-div 11944  df-nn 12290  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12550  df-xnn0 12622  df-z 12636  df-uz 12900  df-rp 13054  df-fz 13564  df-fzo 13708  df-fl 13839  df-mod 13917  df-hash 14376  df-word 14559  df-concat 14615  df-substr 14685  df-pfx 14715  df-csh 14833  df-struct 17189  df-sets 17206  df-slot 17224  df-ndx 17236  df-base 17254  df-ress 17283  df-plusg 17319  df-tset 17325  df-0g 17496  df-mgm 18673  df-sgrp 18752  df-mnd 18768  df-submnd 18814  df-efmnd 18899  df-grp 18971  df-minusg 18972  df-sbg 18973  df-symg 19406  df-tocyc 33092
This theorem is referenced by:  cyc3conja  33142
  Copyright terms: Public domain W3C validator