cycpmconjs - Mathbox for Thierry Arnoux

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Theorem cycpmconjs 33128

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 33127	. 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 33127	. . . . 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 6794	. . . . . . . . 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 6805	. . . . . . . 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 19288	. . . . . . . 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 7384	. . . . . . . 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 7387	. . . . . . 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 2740	. . . . . 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 2767	. . . . . . . . 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 5815	. . . . . . . 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 5816	. . . . . . 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 6226	. . . . . . . . 9 ⊢ ((𝑞 ∘ (^◡𝑞 ∘ 𝑄)) ∘ (𝑞 ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)))
34		coass 6226	. . . . . . . . . 10 ⊢ ((𝑞 ∘ ^◡𝑞) ∘ 𝑄) = (𝑞 ∘ (^◡𝑞 ∘ 𝑄))
35	34	coeq1i 5813	. . . . . . . . 9 ⊢ (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)) = ((𝑞 ∘ (^◡𝑞 ∘ 𝑄)) ∘ (𝑞 ∘ ^◡𝑞))
36		coass 6226	. . . . . . . . . 10 ⊢ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞) = ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞))
37	36	coeq2i 5814	. . . . . . . . 9 ⊢ (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)))
38	33, 35, 37	3eqtr4ri 2763	. . . . . . . 8 ⊢ (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞))
39		f1ococnv2 6809	. . . . . . . . . . . . 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 5815	. . . . . . . . . . 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 33125	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) → 𝐶 ⊆ 𝐵)
43	9, 8, 42	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
44	43, 10	sseldd 3944	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
45	2, 5	elsymgbas 19288	. . . . . . . . . . . . . . 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 6782	. . . . . . . . . . . . 13 ⊢ (𝑄:𝐷–1-1-onto→𝐷 → 𝑄:𝐷⟶𝐷)
49		fcoi2 6717	. . . . . . . . . . . . 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 2764	. . . . . . . . . 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 5818	. . . . . . . . 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 6716	. . . . . . . . . . 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 2764	. . . . . . . 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 2776	. . . . . . 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 6226	. . . . . . . . 9 ⊢ ((𝑞 ∘ (^◡𝑡 ∘ 𝑇)) ∘ (𝑡 ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
60		coass 6226	. . . . . . . . . 10 ⊢ ((𝑞 ∘ ^◡𝑡) ∘ 𝑇) = (𝑞 ∘ (^◡𝑡 ∘ 𝑇))
61	60	coeq1i 5813	. . . . . . . . 9 ⊢ (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = ((𝑞 ∘ (^◡𝑡 ∘ 𝑇)) ∘ (𝑡 ∘ ^◡𝑞))
62		coass 6226	. . . . . . . . . 10 ⊢ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞) = ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞))
63	62	coeq2i 5814	. . . . . . . . 9 ⊢ (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
64	59, 61, 63	3eqtr4i 2762	. . . . . . . 8 ⊢ (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞))
65	43, 12	sseldd 3944	. . . . . . . . . . . 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 19298	. . . . . . . . . . 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 7384	. . . . . . . . 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 19314	. . . . . . . . . . . . . 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 18855	. . . . . . . . . . . 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 2829	. . . . . . . . . 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 33059	. . . . . . . . . 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 5839	. . . . . . . . . . . 12 ⊢ ^◡(𝑞 ∘ ^◡𝑡) = (^◡^◡𝑡 ∘ ^◡𝑞)
79		f1orel 6785	. . . . . . . . . . . . . 14 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → Rel 𝑡)
80		dfrel2 6150	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑡 ↔ ^◡^◡𝑡 = 𝑡)
81	79, 80	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡^◡𝑡 = 𝑡)
82	81	coeq1d 5815	. . . . . . . . . . . 12 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → (^◡^◡𝑡 ∘ ^◡𝑞) = (𝑡 ∘ ^◡𝑞))
83	78, 82	eqtrid 2776	. . . . . . . . . . 11 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡(𝑞 ∘ ^◡𝑡) = (𝑡 ∘ ^◡𝑞))
84	83	coeq2d 5816	. . . . . . . . . 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 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 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = (((𝑞 ∘ ^◡𝑡) + 𝑇) − (𝑞 ∘ ^◡𝑡)))
87	64, 86	eqtr3id 2778	. . . . . . 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 2772	. . . . . 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 3587	. . . . 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 1935	. . 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 1935	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ∃wrex 3053 ∪ cun 3909 ⊆ wss 3911 {csn 4585 I cid 5525 ^◡ccnv 5630 ↾ cres 5633 “ cima 5634 ∘ ccom 5635 Rel wrel 5636 ⟶wf 6495 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 Fincfn 8895 0cc0 11044 1c1 11045 ...cfz 13444 ..^cfzo 13591 ♯chash 14271 cyclShift ccsh 14729 Basecbs 17155 +_gcplusg 17196 Grpcgrp 18847 -_gcsg 18849 SymGrpcsymg 19283 toCycctocyc 33078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-uz 12770 df-rp 12928 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-hash 14272 df-word 14455 df-concat 14512 df-substr 14582 df-pfx 14612 df-csh 14730 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-tset 17215 df-0g 17380 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-efmnd 18778 df-grp 18850 df-minusg 18851 df-sbg 18852 df-symg 19284 df-tocyc 33079

This theorem is referenced by: cyc3conja 33129

W3C validator