cycpmconjs - Mathbox for Thierry Arnoux

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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	⊢ + = (+_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 33150	. 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 33150	. . . . 5 ⊢ (𝜑 → ∃𝑡(𝑡:(0..^𝑁)–1-1-onto→𝐷 ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
14	13	ad2antrr 725	. . . 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 731	. . . . . . 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 6876	. . . . . . . . 9 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡𝑡:𝐷–1-1-onto→(0..^𝑁))
18	17	ad2antlr 726	. . . . . . . 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 6887	. . . . . . . 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 583	. . . . . . 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 19417	. . . . . . . 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 583	. . . . . 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 7465	. . . . . . . 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 7468	. . . . . . 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 2751	. . . . . 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 2783	. . . . . . . . 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 5886	. . . . . . . 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 5887	. . . . . . 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 6298	. . . . . . . . 9 ⊢ ((𝑞 ∘ (^◡𝑞 ∘ 𝑄)) ∘ (𝑞 ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)))
34		coass 6298	. . . . . . . . . 10 ⊢ ((𝑞 ∘ ^◡𝑞) ∘ 𝑄) = (𝑞 ∘ (^◡𝑞 ∘ 𝑄))
35	34	coeq1i 5884	. . . . . . . . 9 ⊢ (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)) = ((𝑞 ∘ (^◡𝑞 ∘ 𝑄)) ∘ (𝑞 ∘ ^◡𝑞))
36		coass 6298	. . . . . . . . . 10 ⊢ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞) = ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞))
37	36	coeq2i 5885	. . . . . . . . 9 ⊢ (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)))
38	33, 35, 37	3eqtr4ri 2779	. . . . . . . 8 ⊢ (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞))
39		f1ococnv2 6891	. . . . . . . . . . . . 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 5886	. . . . . . . . . . 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 33148	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) → 𝐶 ⊆ 𝐵)
43	9, 8, 42	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
44	43, 10	sseldd 4009	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
45	2, 5	elsymgbas 19417	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄:𝐷–1-1-onto→𝐷)
48		f1of 6864	. . . . . . . . . . . . 13 ⊢ (𝑄:𝐷–1-1-onto→𝐷 → 𝑄:𝐷⟶𝐷)
49		fcoi2 6798	. . . . . . . . . . . . 13 ⊢ (𝑄:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
50	47, 48, 49	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
51	50	ad4antr 731	. . . . . . . . . . 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 2780	. . . . . . . . . 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 5889	. . . . . . . . 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 6797	. . . . . . . . . . 11 ⊢ (𝑄:𝐷⟶𝐷 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
55	47, 48, 54	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
56	55	ad4antr 731	. . . . . . . . 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 2780	. . . . . . . 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 2792	. . . . . . 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 6298	. . . . . . . . 9 ⊢ ((𝑞 ∘ (^◡𝑡 ∘ 𝑇)) ∘ (𝑡 ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
60		coass 6298	. . . . . . . . . 10 ⊢ ((𝑞 ∘ ^◡𝑡) ∘ 𝑇) = (𝑞 ∘ (^◡𝑡 ∘ 𝑇))
61	60	coeq1i 5884	. . . . . . . . 9 ⊢ (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = ((𝑞 ∘ (^◡𝑡 ∘ 𝑇)) ∘ (𝑡 ∘ ^◡𝑞))
62		coass 6298	. . . . . . . . . 10 ⊢ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞) = ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞))
63	62	coeq2i 5885	. . . . . . . . 9 ⊢ (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
64	59, 61, 63	3eqtr4i 2778	. . . . . . . 8 ⊢ (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞))
65	43, 12	sseldd 4009	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
66	65	ad4antr 731	. . . . . . . . . . 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 19427	. . . . . . . . . . 11 ⊢ (((𝑞 ∘ ^◡𝑡) ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → ((𝑞 ∘ ^◡𝑡) + 𝑇) = ((𝑞 ∘ ^◡𝑡) ∘ 𝑇))
68	23, 66, 67	syl2anc 583	. . . . . . . . . 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 7465	. . . . . . . . 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 19444	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
71	9, 70	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ Grp)
72	71	ad4antr 731	. . . . . . . . . . . 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 18983	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Grp ∧ (𝑞 ∘ ^◡𝑡) ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → ((𝑞 ∘ ^◡𝑡) + 𝑇) ∈ 𝐵)
74	72, 23, 66, 73	syl3anc 1371	. . . . . . . . . . 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 2845	. . . . . . . . . 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 33082	. . . . . . . . . 10 ⊢ ((((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∈ 𝐵 ∧ (𝑞 ∘ ^◡𝑡) ∈ 𝐵) → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) − (𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ ^◡(𝑞 ∘ ^◡𝑡)))
77	75, 23, 76	syl2anc 583	. . . . . . . . 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 5910	. . . . . . . . . . . 12 ⊢ ^◡(𝑞 ∘ ^◡𝑡) = (^◡^◡𝑡 ∘ ^◡𝑞)
79		f1orel 6867	. . . . . . . . . . . . . 14 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → Rel 𝑡)
80		dfrel2 6222	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑡 ↔ ^◡^◡𝑡 = 𝑡)
81	79, 80	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡^◡𝑡 = 𝑡)
82	81	coeq1d 5886	. . . . . . . . . . . 12 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → (^◡^◡𝑡 ∘ ^◡𝑞) = (𝑡 ∘ ^◡𝑞))
83	78, 82	eqtrid 2792	. . . . . . . . . . 11 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡(𝑞 ∘ ^◡𝑡) = (𝑡 ∘ ^◡𝑞))
84	83	coeq2d 5887	. . . . . . . . . 10 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ ^◡(𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
85	84	ad2antlr 726	. . . . . . . . 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 2785	. . . . . . . 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 2794	. . . . . . 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 2788	. . . . . 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 3637	. . . . 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 1934	. . 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 1934	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 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-onto→wf1o 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

W3C validator