cycpmconjs - Mathbox for Thierry Arnoux

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Theorem cycpmconjs 32934

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 32933	. 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 32933	. . . . 5 ⊢ (𝜑 → ∃𝑡(𝑡:(0..^𝑁)–1-1-onto→𝐷 ∧ ((^◡𝑡 ∘ 𝑇) ∘ 𝑡) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
14	13	ad2antrr 724	. . . 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 730	. . . . . . 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 782	. . . . . . . 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 6848	. . . . . . . . 9 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡𝑡:𝐷–1-1-onto→(0..^𝑁))
18	17	ad2antlr 725	. . . . . . . 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 6859	. . . . . . . 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 582	. . . . . . 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 19332	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ((𝑞 ∘ ^◡𝑡) ∈ 𝐵 ↔ (𝑞 ∘ ^◡𝑡):𝐷–1-1-onto→𝐷))
22	21	biimpar 476	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ (𝑞 ∘ ^◡𝑡):𝐷–1-1-onto→𝐷) → (𝑞 ∘ ^◡𝑡) ∈ 𝐵)
23	15, 20, 22	syl2anc 582	. . . . . 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 483	. . . . . . . . 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 7432	. . . . . . . 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 7435	. . . . . . 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 2736	. . . . . 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 774	. . . . . . . . . 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 483	. . . . . . . . . 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 2768	. . . . . . . . 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 5863	. . . . . . . 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 5864	. . . . . . 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 6269	. . . . . . . . 9 ⊢ ((𝑞 ∘ (^◡𝑞 ∘ 𝑄)) ∘ (𝑞 ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)))
34		coass 6269	. . . . . . . . . 10 ⊢ ((𝑞 ∘ ^◡𝑞) ∘ 𝑄) = (𝑞 ∘ (^◡𝑞 ∘ 𝑄))
35	34	coeq1i 5861	. . . . . . . . 9 ⊢ (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)) = ((𝑞 ∘ (^◡𝑞 ∘ 𝑄)) ∘ (𝑞 ∘ ^◡𝑞))
36		coass 6269	. . . . . . . . . 10 ⊢ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞) = ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞))
37	36	coeq2i 5862	. . . . . . . . 9 ⊢ (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑞 ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞)))
38	33, 35, 37	3eqtr4ri 2764	. . . . . . . 8 ⊢ (𝑞 ∘ (((^◡𝑞 ∘ 𝑄) ∘ 𝑞) ∘ ^◡𝑞)) = (((𝑞 ∘ ^◡𝑞) ∘ 𝑄) ∘ (𝑞 ∘ ^◡𝑞))
39		f1ococnv2 6863	. . . . . . . . . . . . 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 5863	. . . . . . . . . . 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 32931	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) → 𝐶 ⊆ 𝐵)
43	9, 8, 42	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
44	43, 10	sseldd 3978	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
45	2, 5	elsymgbas 19332	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ 𝐵 ↔ 𝑄:𝐷–1-1-onto→𝐷))
46	45	biimpa 475	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐵) → 𝑄:𝐷–1-1-onto→𝐷)
47	9, 44, 46	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄:𝐷–1-1-onto→𝐷)
48		f1of 6836	. . . . . . . . . . . . 13 ⊢ (𝑄:𝐷–1-1-onto→𝐷 → 𝑄:𝐷⟶𝐷)
49		fcoi2 6770	. . . . . . . . . . . . 13 ⊢ (𝑄:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
50	47, 48, 49	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (( I ↾ 𝐷) ∘ 𝑄) = 𝑄)
51	50	ad4antr 730	. . . . . . . . . . 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 2765	. . . . . . . . . 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 5866	. . . . . . . . 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 6769	. . . . . . . . . . 11 ⊢ (𝑄:𝐷⟶𝐷 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
55	47, 48, 54	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∘ ( I ↾ 𝐷)) = 𝑄)
56	55	ad4antr 730	. . . . . . . . 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 2765	. . . . . . . 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 2777	. . . . . . 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 6269	. . . . . . . . 9 ⊢ ((𝑞 ∘ (^◡𝑡 ∘ 𝑇)) ∘ (𝑡 ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
60		coass 6269	. . . . . . . . . 10 ⊢ ((𝑞 ∘ ^◡𝑡) ∘ 𝑇) = (𝑞 ∘ (^◡𝑡 ∘ 𝑇))
61	60	coeq1i 5861	. . . . . . . . 9 ⊢ (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = ((𝑞 ∘ (^◡𝑡 ∘ 𝑇)) ∘ (𝑡 ∘ ^◡𝑞))
62		coass 6269	. . . . . . . . . 10 ⊢ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞) = ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞))
63	62	coeq2i 5862	. . . . . . . . 9 ⊢ (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞)) = (𝑞 ∘ ((^◡𝑡 ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
64	59, 61, 63	3eqtr4i 2763	. . . . . . . 8 ⊢ (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)) = (𝑞 ∘ (((^◡𝑡 ∘ 𝑇) ∘ 𝑡) ∘ ^◡𝑞))
65	43, 12	sseldd 3978	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
66	65	ad4antr 730	. . . . . . . . . . 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 19342	. . . . . . . . . . 11 ⊢ (((𝑞 ∘ ^◡𝑡) ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → ((𝑞 ∘ ^◡𝑡) + 𝑇) = ((𝑞 ∘ ^◡𝑡) ∘ 𝑇))
68	23, 66, 67	syl2anc 582	. . . . . . . . . 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 7432	. . . . . . . . 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 19359	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
71	9, 70	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ Grp)
72	71	ad4antr 730	. . . . . . . . . . . 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 18902	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Grp ∧ (𝑞 ∘ ^◡𝑡) ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → ((𝑞 ∘ ^◡𝑡) + 𝑇) ∈ 𝐵)
74	72, 23, 66, 73	syl3anc 1368	. . . . . . . . . . 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 2826	. . . . . . . . . 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 32867	. . . . . . . . . 10 ⊢ ((((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∈ 𝐵 ∧ (𝑞 ∘ ^◡𝑡) ∈ 𝐵) → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) − (𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ ^◡(𝑞 ∘ ^◡𝑡)))
77	75, 23, 76	syl2anc 582	. . . . . . . . 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 5887	. . . . . . . . . . . 12 ⊢ ^◡(𝑞 ∘ ^◡𝑡) = (^◡^◡𝑡 ∘ ^◡𝑞)
79		f1orel 6839	. . . . . . . . . . . . . 14 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → Rel 𝑡)
80		dfrel2 6193	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑡 ↔ ^◡^◡𝑡 = 𝑡)
81	79, 80	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡^◡𝑡 = 𝑡)
82	81	coeq1d 5863	. . . . . . . . . . . 12 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → (^◡^◡𝑡 ∘ ^◡𝑞) = (𝑡 ∘ ^◡𝑞))
83	78, 82	eqtrid 2777	. . . . . . . . . . 11 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → ^◡(𝑞 ∘ ^◡𝑡) = (𝑡 ∘ ^◡𝑞))
84	83	coeq2d 5864	. . . . . . . . . 10 ⊢ (𝑡:(0..^𝑁)–1-1-onto→𝐷 → (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ ^◡(𝑞 ∘ ^◡𝑡)) = (((𝑞 ∘ ^◡𝑡) ∘ 𝑇) ∘ (𝑡 ∘ ^◡𝑞)))
85	84	ad2antlr 725	. . . . . . . . 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 2770	. . . . . . . 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 2779	. . . . . . 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 2773	. . . . . 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 3609	. . . . 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 465	. . . 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 1930	. . 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 465	. 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 1930	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃wrex 3060 ∪ cun 3943 ⊆ wss 3945 {csn 4629 I cid 5574 ^◡ccnv 5676 ↾ cres 5679 “ cima 5680 ∘ ccom 5681 Rel wrel 5682 ⟶wf 6543 –1-1-onto→wf1o 6546 ‘cfv 6547 (class class class)co 7417 Fincfn 8962 0cc0 11138 1c1 11139 ...cfz 13516 ..^cfzo 13659 ♯chash 14321 cyclShift ccsh 14770 Basecbs 17179 +_gcplusg 17232 Grpcgrp 18894 -_gcsg 18896 SymGrpcsymg 19325 toCycctocyc 32884

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-hash 14322 df-word 14497 df-concat 14553 df-substr 14623 df-pfx 14653 df-csh 14771 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-tset 17251 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-efmnd 18825 df-grp 18897 df-minusg 18898 df-sbg 18899 df-symg 19326 df-tocyc 32885

This theorem is referenced by: cyc3conja 32935

W3C validator