cycpmconjs - Mathbox for Thierry Arnoux

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Theorem cycpmconjs 33187

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	⊢ + = (+_g‘𝑆)
cycpmconjs.l	⊢ − = (-_g‘𝑆)
cycpmconjs.p	⊢ (𝜑 → 𝑃 ∈ (0...𝑁))
cycpmconjs.d	⊢ (𝜑 → 𝐷 ∈ Fin)
cycpmconjs.q	⊢ (𝜑 → 𝑄 ∈ 𝐶)
cycpmconjs.t	⊢ (𝜑 → 𝑇 ∈ 𝐶)

**Assertion**
Ref	Expression
cycpmconjs	⊢ (𝜑 → ∃𝑝 ∈ 𝐵 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.

Step	Hyp	Ref	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 ⊢ + = (+_g‘𝑆)
7		cycpmconjs.l	. . 3 ⊢ − = (-_g‘𝑆)
8		cycpmconjs.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ (0...𝑁))
9		cycpmconjs.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Fin)
10		cycpmconjs.q	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐶)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cycpmconjslem2 33186	. 2 ⊢ (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto→𝐷 ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
12		cycpmconjs.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐶)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 12	cycpmconjslem2 33186	. . . . 5 ⊢ (𝜑 → ∃𝑡(𝑡:(0..^𝑁)–1-1-onto→𝐷 ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
14	13	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ∃𝑡(𝑡:(0..^𝑁)–1-1-onto→𝐷 ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
15	9	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝐷 ∈ Fin)
16		simp-4r 783	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑞:(0..^𝑁)–1-1-onto→𝐷)
17		f1ocnv 6784	. . . . . . . . 9 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡𝑡:𝐷–1-1-onto→(0..^𝑁))
18	17	ad2antlr 727	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ^◡𝑡:𝐷–1-1-onto→(0..^𝑁))
19		f1oco 6795	. . . . . . . 8 ⊢ ((𝑞:(0..^𝑁)–1-1-onto→𝐷 ∧ ^◡𝑡:𝐷–1-1-onto→(0..^𝑁)) → (𝑞 ∘ ^◡𝑡):𝐷–1-1-onto→𝐷)
20	16, 18, 19	syl2anc 584	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ ^◡𝑡):𝐷–1-1-onto→𝐷)
21	2, 5	elsymgbas 19301	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ((𝑞 ∘ ^◡𝑡) ∈ 𝐵 ↔ (𝑞 ∘ ^◡𝑡):𝐷–1-1-onto→𝐷))
22	21	biimpar 477	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ (𝑞 ∘ ^◡𝑡):𝐷–1-1-onto→𝐷) → (𝑞 ∘ ^◡𝑡) ∈ 𝐵)
23	15, 20, 22	syl2anc 584	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ ^◡𝑡) ∈ 𝐵)
24		simpr 484	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑝 = (𝑞 ∘ ^◡𝑡)) → 𝑝 = (𝑞 ∘ ^◡𝑡))
25	24	oveq1d 7371	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑝 = (𝑞 ∘ ^◡𝑡)) → (𝑝 + 𝑇) = ((𝑞 ∘ ^◡𝑡) + 𝑇))
26	25, 24	oveq12d 7374	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑝 = (𝑞 ∘ ^◡𝑡)) → ((𝑝 + 𝑇) − 𝑝) = (((𝑞 ∘ ^◡𝑡) + 𝑇) − (𝑞 ∘ ^◡𝑡)))
27	26	eqeq2d 2745	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑝 = (𝑞 ∘ ^◡𝑡)) → (𝑄 = ((𝑝 + 𝑇) − 𝑝) ↔ 𝑄 = (((𝑞 ∘ ^◡𝑡) + 𝑇) − (𝑞 ∘ ^◡𝑡))))
28		simpllr 775	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))
29		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))
30	28, 29	eqtr4d 2772	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((^◡𝑡 ∘ 𝑇) ∘ 𝑡))
31	30	coeq1d 5808	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞) = (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞))
32	31	coeq2d 5809	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞)))
33		coass 6222	. . . . . . . . 9 ⊢ ((𝑞 ∘ (^◡𝑞 ∘ 𝑄)) ∘ (𝑞 ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)))
34		coass 6222	. . . . . . . . . 10 ⊢ ((𝑞 ∘ ^◡𝑞) ∘ 𝑄) = (𝑞 ∘ (^◡𝑞 ∘ 𝑄))
35	34	coeq1i 5806	. . . . . . . . 9 ⊢ (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)) = ((𝑞 ∘ (^◡𝑞 ∘ 𝑄)) ∘ (𝑞 ∘ ^◡𝑞))
36		coass 6222	. . . . . . . . . 10 ⊢ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞) = ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞))
37	36	coeq2i 5807	. . . . . . . . 9 ⊢ (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)))
38	33, 35, 37	3eqtr4ri 2768	. . . . . . . 8 ⊢ (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞))
39		f1ococnv2 6799	. . . . . . . . . . . . 13 ⊢ (𝑞:(0..^𝑁)–1-1-onto→𝐷 → (𝑞 ∘ ^◡𝑞) = ( I ↾ 𝐷))
40	16, 39	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ ^◡𝑞) = ( I ↾ 𝐷))
41	40	coeq1d 5808	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞 ∘ ^◡𝑞) ∘ 𝑄) = (( I ↾ 𝐷) ∘ 𝑄))
42	1, 2, 3, 4, 5	cycpmgcl 33184	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) → 𝐶 ⊆ 𝐵)
43	9, 8, 42	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
44	43, 10	sseldd 3932	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
45	2, 5	elsymgbas 19301	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ 𝐵 ↔ 𝑄:𝐷–1-1-onto→𝐷))
46	45	biimpa 476	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐵) → 𝑄:𝐷–1-1-onto→𝐷)
47	9, 44, 46	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄:𝐷–1-1-onto→𝐷)
48		f1of 6772	. . . . . . . . . . . . 13 ⊢ (𝑄:𝐷–1-1-onto→𝐷 → 𝑄:𝐷⟶𝐷)
49		fcoi2 6707	. . . . . . . . . . . . 13 ⊢ (𝑄:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
50	47, 48, 49	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
51	50	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
52	41, 51	eqtrd 2769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞 ∘ ^◡𝑞) ∘ 𝑄) = 𝑄)
53	52, 40	coeq12d 5811	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)) = (𝑄 ∘ ( I ↾ 𝐷)))
54		fcoi1 6706	. . . . . . . . . . 11 ⊢ (𝑄:𝐷⟶𝐷 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
55	47, 48, 54	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
56	55	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
57	53, 56	eqtrd 2769	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)) = 𝑄)
58	38, 57	eqtrid 2781	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = 𝑄)
59		coass 6222	. . . . . . . . 9 ⊢ ((𝑞 ∘ (^◡𝑡 ∘ 𝑇)) ∘ (𝑡 ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
60		coass 6222	. . . . . . . . . 10 ⊢ ((𝑞 ∘ ^◡𝑡) ∘ 𝑇) = (𝑞 ∘ (^◡𝑡 ∘ 𝑇))
61	60	coeq1i 5806	. . . . . . . . 9 ⊢ (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = ((𝑞 ∘ (^◡𝑡 ∘ 𝑇)) ∘ (𝑡 ∘ ^◡𝑞))
62		coass 6222	. . . . . . . . . 10 ⊢ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞) = ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞))
63	62	coeq2i 5807	. . . . . . . . 9 ⊢ (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
64	59, 61, 63	3eqtr4i 2767	. . . . . . . 8 ⊢ (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞))
65	43, 12	sseldd 3932	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
66	65	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑇 ∈ 𝐵)
67	2, 5, 6	symgov 19311	. . . . . . . . . . 11 ⊢ (((𝑞 ∘ ^◡𝑡) ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → ((𝑞 ∘ ^◡𝑡) + 𝑇) = ((𝑞 ∘ ^◡𝑡) ∘ 𝑇))
68	23, 66, 67	syl2anc 584	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞 ∘ ^◡𝑡) + 𝑇) = ((𝑞 ∘ ^◡𝑡) ∘ 𝑇))
69	68	oveq1d 7371	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞 ∘ ^◡𝑡) + 𝑇) − (𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) − (𝑞 ∘ ^◡𝑡)))
70	2	symggrp 19327	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
71	9, 70	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ Grp)
72	71	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑆 ∈ Grp)
73	5, 6	grpcl 18869	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Grp ∧ (𝑞 ∘ ^◡𝑡) ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → ((𝑞 ∘ ^◡𝑡) + 𝑇) ∈ 𝐵)
74	72, 23, 66, 73	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞 ∘ ^◡𝑡) + 𝑇) ∈ 𝐵)
75	68, 74	eqeltrrd 2835	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∈ 𝐵)
76	2, 5, 7	symgsubg 33118	. . . . . . . . . 10 ⊢ ((((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∈ 𝐵 ∧ (𝑞 ∘ ^◡𝑡) ∈ 𝐵) → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) − (𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ ^◡(𝑞 ∘ ^◡𝑡)))
77	75, 23, 76	syl2anc 584	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) − (𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ ^◡(𝑞 ∘ ^◡𝑡)))
78		cnvco 5832	. . . . . . . . . . . 12 ⊢ ^◡(𝑞 ∘ ^◡𝑡) = (^◡^◡𝑡 ∘ ^◡𝑞)
79		f1orel 6775	. . . . . . . . . . . . . 14 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → Rel 𝑡)
80		dfrel2 6145	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑡 ↔ ^◡^◡𝑡 = 𝑡)
81	79, 80	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡^◡𝑡 = 𝑡)
82	81	coeq1d 5808	. . . . . . . . . . . 12 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → (^◡^◡𝑡 ∘ ^◡𝑞) = (𝑡 ∘ ^◡𝑞))
83	78, 82	eqtrid 2781	. . . . . . . . . . 11 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡(𝑞 ∘ ^◡𝑡) = (𝑡 ∘ ^◡𝑞))
84	83	coeq2d 5809	. . . . . . . . . 10 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ ^◡(𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
85	84	ad2antlr 727	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ ^◡(𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
86	69, 77, 85	3eqtrrd 2774	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = (((𝑞 ∘ ^◡𝑡) + 𝑇) − (𝑞 ∘ ^◡𝑡)))
87	64, 86	eqtr3id 2783	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞)) = (((𝑞 ∘ ^◡𝑡) + 𝑇) − (𝑞 ∘ ^◡𝑡)))
88	32, 58, 87	3eqtr3d 2777	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → 𝑄 = (((𝑞 ∘ ^◡𝑡) + 𝑇) − (𝑞 ∘ ^◡𝑡)))
89	23, 27, 88	rspcedvd 3576	. . . . 5 ⊢ (((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ 𝑡:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝))
90	89	anasss 466	. . . 4 ⊢ ((((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) ∧ (𝑡:(0..^𝑁)–1-1-onto→𝐷 ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))) → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝))
91	14, 90	exlimddv 1936	. . 3 ⊢ (((𝜑 ∧ 𝑞:(0..^𝑁)–1-1-onto→𝐷) ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))) → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝))
92	91	anasss 466	. 2 ⊢ ((𝜑 ∧ (𝑞:(0..^𝑁)–1-1-onto→𝐷 ∧ ((^◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))) → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝))
93	11, 92	exlimddv 1936	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ∃wrex 3058 ∪ cun 3897 ⊆ wss 3899 {csn 4578 I cid 5516 ^◡ccnv 5621 ↾ cres 5624 “ cima 5625 ∘ ccom 5626 Rel wrel 5627 ⟶wf 6486 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7356 Fincfn 8881 0cc0 11024 1c1 11025 ...cfz 13421 ..^cfzo 13568 ♯chash 14251 cyclShift ccsh 14709 Basecbs 17134 +_gcplusg 17175 Grpcgrp 18861 -_gcsg 18863 SymGrpcsymg 19296 toCycctocyc 33137

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-uz 12750 df-rp 12904 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-hash 14252 df-word 14435 df-concat 14492 df-substr 14563 df-pfx 14593 df-csh 14710 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-tset 17194 df-0g 17359 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-efmnd 18792 df-grp 18864 df-minusg 18865 df-sbg 18866 df-symg 19297 df-tocyc 33138

This theorem is referenced by: cyc3conja 33188

W3C validator