cycpmconjv - Mathbox for Thierry Arnoux

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Theorem cycpmconjv 33276

Description: A formula for computing conjugacy classes of cyclic permutations. Formula in property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 9-Oct-2023.)

**Hypotheses**
Ref	Expression
cycpmconjv.s	⊢ 𝑆 = (SymGrp‘𝐷)
cycpmconjv.m	⊢ 𝑀 = (toCyc‘𝐷)
cycpmconjv.p	⊢ + = (+_g‘𝑆)
cycpmconjv.l	⊢ − = (-_g‘𝑆)
cycpmconjv.b	⊢ 𝐵 = (Base‘𝑆)

**Assertion**
Ref	Expression
cycpmconjv	⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (𝑀‘(𝐺 ∘ 𝑊)))

Proof of Theorem cycpmconjv Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpmconjv.s	. . . . . . 7 ⊢ 𝑆 = (SymGrp‘𝐷)
2		cycpmconjv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
3	1, 2	symgbasf1o 19391	. . . . . 6 ⊢ (𝐺 ∈ 𝐵 → 𝐺:𝐷–1-1-onto→𝐷)
4	3	3ad2ant2 1143	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝐺:𝐷–1-1-onto→𝐷)
5		simp3 1147	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊 ∈ dom 𝑀)
6		cycpmconjv.m	. . . . . . . . . . . . 13 ⊢ 𝑀 = (toCyc‘𝐷)
7	6, 1, 2	tocycf 33251	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶𝐵)
8	7	3ad2ant1 1142	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶𝐵)
9	8	fdmd 6691	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	5, 9	eleqtrd 2858	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
11		id 22	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
12		dmeq 5872	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
13		eqidd 2757	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
14	11, 12, 13	f1eq123d 6787	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
15	14	elrab 3645	. . . . . . . . 9 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
16	10, 15	sylib 220	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
17	16	simprd 498	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊:dom 𝑊–1-1→𝐷)
18		f1f 6749	. . . . . . 7 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊⟶𝐷)
19	17, 18	syl 17	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊:dom 𝑊⟶𝐷)
20	19	frnd 6689	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ran 𝑊 ⊆ 𝐷)
21	4, 20	cycpmconjvlem 33275	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) = ( I ↾ (𝐷 ∖ ran (𝐺 ↾ ran 𝑊))))
22		rnco 6228	. . . . . 6 ⊢ ran (𝐺 ∘ 𝑊) = ran (𝐺 ↾ ran 𝑊)
23	22	difeq2i 4072	. . . . 5 ⊢ (𝐷 ∖ ran (𝐺 ∘ 𝑊)) = (𝐷 ∖ ran (𝐺 ↾ ran 𝑊))
24	23	reseq2i 5955	. . . 4 ⊢ ( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))) = ( I ↾ (𝐷 ∖ ran (𝐺 ↾ ran 𝑊)))
25	21, 24	eqtr4di 2809	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) = ( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))))
26		coass 6242	. . . . 5 ⊢ ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ (^◡𝑊 ∘ ^◡𝐺))
27		cnvco 5854	. . . . . 6 ⊢ ^◡(𝐺 ∘ 𝑊) = (^◡𝑊 ∘ ^◡𝐺)
28	27	coeq2i 5825	. . . . 5 ⊢ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡(𝐺 ∘ 𝑊)) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ (^◡𝑊 ∘ ^◡𝐺))
29	26, 28	eqtr4i 2782	. . . 4 ⊢ ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡(𝐺 ∘ 𝑊))
30	29	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡(𝐺 ∘ 𝑊)))
31	25, 30	uneq12d 4117	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) ∪ ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺)) = (( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡(𝐺 ∘ 𝑊))))
32		simp2 1146	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝐺 ∈ 𝐵)
33	8, 10	ffvelcdmd 7055	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘𝑊) ∈ 𝐵)
34		cycpmconjv.p	. . . . . . . 8 ⊢ + = (+_g‘𝑆)
35	1, 2, 34	symgcl 19401	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑀‘𝑊) ∈ 𝐵) → (𝐺 + (𝑀‘𝑊)) ∈ 𝐵)
36	32, 33, 35	syl2anc 592	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) ∈ 𝐵)
37		eqid 2756	. . . . . . 7 ⊢ (inv_g‘𝑆) = (inv_g‘𝑆)
38		cycpmconjv.l	. . . . . . 7 ⊢ − = (-_g‘𝑆)
39	2, 34, 37, 38	grpsubval 19003	. . . . . 6 ⊢ (((𝐺 + (𝑀‘𝑊)) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = ((𝐺 + (𝑀‘𝑊)) + ((inv_g‘𝑆)‘𝐺)))
40	36, 32, 39	syl2anc 592	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = ((𝐺 + (𝑀‘𝑊)) + ((inv_g‘𝑆)‘𝐺)))
41	1, 2, 37	symginv 19418	. . . . . . 7 ⊢ (𝐺 ∈ 𝐵 → ((inv_g‘𝑆)‘𝐺) = ^◡𝐺)
42	41	3ad2ant2 1143	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((inv_g‘𝑆)‘𝐺) = ^◡𝐺)
43	42	oveq2d 7401	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) + ((inv_g‘𝑆)‘𝐺)) = ((𝐺 + (𝑀‘𝑊)) + ^◡𝐺))
44		simp1 1145	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝐷 ∈ 𝑉)
45		f1ocnv 6808	. . . . . . . 8 ⊢ (𝐺:𝐷–1-1-onto→𝐷 → ^◡𝐺:𝐷–1-1-onto→𝐷)
46	4, 45	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ^◡𝐺:𝐷–1-1-onto→𝐷)
47	1, 2	elsymgbas 19390	. . . . . . . 8 ⊢ (𝐷 ∈ 𝑉 → (^◡𝐺 ∈ 𝐵 ↔ ^◡𝐺:𝐷–1-1-onto→𝐷))
48	47	biimpar 480	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ ^◡𝐺:𝐷–1-1-onto→𝐷) → ^◡𝐺 ∈ 𝐵)
49	44, 46, 48	syl2anc 592	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ^◡𝐺 ∈ 𝐵)
50	1, 2, 34	symgov 19400	. . . . . 6 ⊢ (((𝐺 + (𝑀‘𝑊)) ∈ 𝐵 ∧ ^◡𝐺 ∈ 𝐵) → ((𝐺 + (𝑀‘𝑊)) + ^◡𝐺) = ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺))
51	36, 49, 50	syl2anc 592	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) + ^◡𝐺) = ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺))
52	40, 43, 51	3eqtrd 2795	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺))
53	1, 2, 34	symgov 19400	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑀‘𝑊) ∈ 𝐵) → (𝐺 + (𝑀‘𝑊)) = (𝐺 ∘ (𝑀‘𝑊)))
54	32, 33, 53	syl2anc 592	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = (𝐺 ∘ (𝑀‘𝑊)))
55	16	simpld 497	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊 ∈ Word 𝐷)
56	6, 44, 55, 17	tocycfv 33243	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)))
57	56	coeq2d 5827	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (𝑀‘𝑊)) = (𝐺 ∘ (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
58		coundi 6223	. . . . . . . 8 ⊢ (𝐺 ∘ (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊))) = ((𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) ∪ (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊)))
59	58	a1i 11	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊))) = ((𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) ∪ (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
60	54, 57, 59	3eqtrd 2795	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = ((𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) ∪ (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
61		coires1 6241	. . . . . . . 8 ⊢ (𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) = (𝐺 ↾ (𝐷 ∖ ran 𝑊))
62	61	a1i 11	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) = (𝐺 ↾ (𝐷 ∖ ran 𝑊)))
63		coass 6242	. . . . . . . 8 ⊢ ((𝐺 ∘ (𝑊 cyclShift 1)) ∘ ^◡𝑊) = (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))
64		1zzd 12592	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 1 ∈ ℤ)
65		f1of 6795	. . . . . . . . . . 11 ⊢ (𝐺:𝐷–1-1-onto→𝐷 → 𝐺:𝐷⟶𝐷)
66	4, 65	syl 17	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝐺:𝐷⟶𝐷)
67		cshco 14839	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ ∧ 𝐺:𝐷⟶𝐷) → (𝐺 ∘ (𝑊 cyclShift 1)) = ((𝐺 ∘ 𝑊) cyclShift 1))
68	55, 64, 66, 67	syl3anc 1386	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (𝑊 cyclShift 1)) = ((𝐺 ∘ 𝑊) cyclShift 1))
69	68	coeq1d 5826	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ∘ (𝑊 cyclShift 1)) ∘ ^◡𝑊) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊))
70	63, 69	eqtr3id 2805	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊))
71	62, 70	uneq12d 4117	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) ∪ (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))) = ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)))
72	60, 71	eqtrd 2791	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)))
73	72	coeq1d 5826	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺) = (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)) ∘ ^◡𝐺))
74	52, 73	eqtrd 2791	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)) ∘ ^◡𝐺))
75		coundir 6224	. . 3 ⊢ (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)) ∘ ^◡𝐺) = (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) ∪ ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺))
76	74, 75	eqtrdi 2807	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) ∪ ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺)))
77		wrdco 14834	. . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐺:𝐷⟶𝐷) → (𝐺 ∘ 𝑊) ∈ Word 𝐷)
78	55, 66, 77	syl2anc 592	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ 𝑊) ∈ Word 𝐷)
79		f1of1 6794	. . . . . 6 ⊢ (𝐺:𝐷–1-1-onto→𝐷 → 𝐺:𝐷–1-1→𝐷)
80	4, 79	syl 17	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝐺:𝐷–1-1→𝐷)
81		f1co 6762	. . . . 5 ⊢ ((𝐺:𝐷–1-1→𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷) → (𝐺 ∘ 𝑊):dom 𝑊–1-1→𝐷)
82	80, 17, 81	syl2anc 592	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ 𝑊):dom 𝑊–1-1→𝐷)
83	66	fdmd 6691	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom 𝐺 = 𝐷)
84	20, 83	sseqtrrd 3968	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ran 𝑊 ⊆ dom 𝐺)
85		dmcosseq 5947	. . . . . 6 ⊢ (ran 𝑊 ⊆ dom 𝐺 → dom (𝐺 ∘ 𝑊) = dom 𝑊)
86	84, 85	syl 17	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom (𝐺 ∘ 𝑊) = dom 𝑊)
87		f1eq2 6745	. . . . 5 ⊢ (dom (𝐺 ∘ 𝑊) = dom 𝑊 → ((𝐺 ∘ 𝑊):dom (𝐺 ∘ 𝑊)–1-1→𝐷 ↔ (𝐺 ∘ 𝑊):dom 𝑊–1-1→𝐷))
88	86, 87	syl 17	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ∘ 𝑊):dom (𝐺 ∘ 𝑊)–1-1→𝐷 ↔ (𝐺 ∘ 𝑊):dom 𝑊–1-1→𝐷))
89	82, 88	mpbird 259	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ 𝑊):dom (𝐺 ∘ 𝑊)–1-1→𝐷)
90	6, 44, 78, 89	tocycfv 33243	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘(𝐺 ∘ 𝑊)) = (( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡(𝐺 ∘ 𝑊))))
91	31, 76, 90	3eqtr4d 2801	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (𝑀‘(𝐺 ∘ 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1095 = wceq 1554 ∈ wcel 2136 {crab 3408 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 I cid 5534 ^◡ccnv 5639 dom cdm 5640 ran crn 5641 ↾ cres 5642 ∘ ccom 5644 ⟶wf 6506 –1-1→wf1 6507 –1-1-onto→wf1o 6509 ‘cfv 6510 (class class class)co 7385 1c1 11064 ℤcz 12558 Word cword 14516 cyclShift ccsh 14791 Basecbs 17221 +_gcplusg 17262 inv_gcminusg 18952 -_gcsg 18953 SymGrpcsymg 19385 toCycctocyc 33240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-pre-sup 11141

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-om 7836 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-er 8666 df-map 8798 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-sup 9378 df-inf 9379 df-card 9887 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-z 12559 df-uz 12830 df-rp 12984 df-fz 13503 df-fzo 13650 df-fl 13792 df-mod 13870 df-hash 14334 df-word 14517 df-concat 14574 df-substr 14645 df-pfx 14675 df-csh 14792 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-tset 17281 df-0g 17446 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-submnd 18794 df-efmnd 18879 df-grp 18954 df-minusg 18955 df-sbg 18956 df-symg 19386 df-tocyc 33241

This theorem is referenced by: (None)

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