cycpmconjv - Mathbox for Thierry Arnoux

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Theorem cycpmconjv 31815

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 19114	. . . . . 6 ⊢ (𝐺 ∈ 𝐵 → 𝐺:𝐷–1-1-onto→𝐷)
4	3	3ad2ant2 1134	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝐺:𝐷–1-1-onto→𝐷)
5		simp3 1138	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊 ∈ dom 𝑀)
6		cycpmconjv.m	. . . . . . . . . . . . 13 ⊢ 𝑀 = (toCyc‘𝐷)
7	6, 1, 2	tocycf 31790	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶𝐵)
8	7	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶𝐵)
9	8	fdmd 6676	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	5, 9	eleqtrd 2840	. . . . . . . . 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 5857	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
13		eqidd 2738	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
14	11, 12, 13	f1eq123d 6773	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
15	14	elrab 3643	. . . . . . . . 9 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
16	10, 15	sylib 217	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
17	16	simprd 496	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊:dom 𝑊–1-1→𝐷)
18		f1f 6735	. . . . . . 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 6673	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ran 𝑊 ⊆ 𝐷)
21	4, 20	cycpmconjvlem 31814	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) = ( I ↾ (𝐷 ∖ ran (𝐺 ↾ ran 𝑊))))
22		rnco 6202	. . . . . 6 ⊢ ran (𝐺 ∘ 𝑊) = ran (𝐺 ↾ ran 𝑊)
23	22	difeq2i 4077	. . . . 5 ⊢ (𝐷 ∖ ran (𝐺 ∘ 𝑊)) = (𝐷 ∖ ran (𝐺 ↾ ran 𝑊))
24	23	reseq2i 5932	. . . 4 ⊢ ( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))) = ( I ↾ (𝐷 ∖ ran (𝐺 ↾ ran 𝑊)))
25	21, 24	eqtr4di 2795	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) = ( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))))
26		coass 6215	. . . . 5 ⊢ ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ (^◡𝑊 ∘ ^◡𝐺))
27		cnvco 5839	. . . . . 6 ⊢ ^◡(𝐺 ∘ 𝑊) = (^◡𝑊 ∘ ^◡𝐺)
28	27	coeq2i 5814	. . . . 5 ⊢ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡(𝐺 ∘ 𝑊)) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ (^◡𝑊 ∘ ^◡𝐺))
29	26, 28	eqtr4i 2768	. . . 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 4122	. 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 1137	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝐺 ∈ 𝐵)
33	8, 10	ffvelcdmd 7032	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘𝑊) ∈ 𝐵)
34		cycpmconjv.p	. . . . . . . 8 ⊢ + = (+_g‘𝑆)
35	1, 2, 34	symgcl 19124	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑀‘𝑊) ∈ 𝐵) → (𝐺 + (𝑀‘𝑊)) ∈ 𝐵)
36	32, 33, 35	syl2anc 584	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) ∈ 𝐵)
37		eqid 2737	. . . . . . 7 ⊢ (inv_g‘𝑆) = (inv_g‘𝑆)
38		cycpmconjv.l	. . . . . . 7 ⊢ − = (-_g‘𝑆)
39	2, 34, 37, 38	grpsubval 18755	. . . . . 6 ⊢ (((𝐺 + (𝑀‘𝑊)) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = ((𝐺 + (𝑀‘𝑊)) + ((inv_g‘𝑆)‘𝐺)))
40	36, 32, 39	syl2anc 584	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = ((𝐺 + (𝑀‘𝑊)) + ((inv_g‘𝑆)‘𝐺)))
41	1, 2, 37	symginv 19142	. . . . . . 7 ⊢ (𝐺 ∈ 𝐵 → ((inv_g‘𝑆)‘𝐺) = ^◡𝐺)
42	41	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((inv_g‘𝑆)‘𝐺) = ^◡𝐺)
43	42	oveq2d 7367	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) + ((inv_g‘𝑆)‘𝐺)) = ((𝐺 + (𝑀‘𝑊)) + ^◡𝐺))
44		simp1 1136	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝐷 ∈ 𝑉)
45		f1ocnv 6793	. . . . . . . 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 19113	. . . . . . . 8 ⊢ (𝐷 ∈ 𝑉 → (^◡𝐺 ∈ 𝐵 ↔ ^◡𝐺:𝐷–1-1-onto→𝐷))
48	47	biimpar 478	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ ^◡𝐺:𝐷–1-1-onto→𝐷) → ^◡𝐺 ∈ 𝐵)
49	44, 46, 48	syl2anc 584	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ^◡𝐺 ∈ 𝐵)
50	1, 2, 34	symgov 19123	. . . . . 6 ⊢ (((𝐺 + (𝑀‘𝑊)) ∈ 𝐵 ∧ ^◡𝐺 ∈ 𝐵) → ((𝐺 + (𝑀‘𝑊)) + ^◡𝐺) = ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺))
51	36, 49, 50	syl2anc 584	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) + ^◡𝐺) = ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺))
52	40, 43, 51	3eqtrd 2781	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺))
53	1, 2, 34	symgov 19123	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑀‘𝑊) ∈ 𝐵) → (𝐺 + (𝑀‘𝑊)) = (𝐺 ∘ (𝑀‘𝑊)))
54	32, 33, 53	syl2anc 584	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = (𝐺 ∘ (𝑀‘𝑊)))
55	16	simpld 495	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊 ∈ Word 𝐷)
56	6, 44, 55, 17	tocycfv 31782	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)))
57	56	coeq2d 5816	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (𝑀‘𝑊)) = (𝐺 ∘ (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
58		coundi 6197	. . . . . . . 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 2781	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = ((𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) ∪ (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
61		coires1 6214	. . . . . . . 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 6215	. . . . . . . 8 ⊢ ((𝐺 ∘ (𝑊 cyclShift 1)) ∘ ^◡𝑊) = (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))
64		1zzd 12492	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 1 ∈ ℤ)
65		f1of 6781	. . . . . . . . . . 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 14682	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ ∧ 𝐺:𝐷⟶𝐷) → (𝐺 ∘ (𝑊 cyclShift 1)) = ((𝐺 ∘ 𝑊) cyclShift 1))
68	55, 64, 66, 67	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (𝑊 cyclShift 1)) = ((𝐺 ∘ 𝑊) cyclShift 1))
69	68	coeq1d 5815	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ∘ (𝑊 cyclShift 1)) ∘ ^◡𝑊) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊))
70	63, 69	eqtr3id 2791	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊))
71	62, 70	uneq12d 4122	. . . . . 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 2777	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)))
73	72	coeq1d 5815	. . . 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 2777	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)) ∘ ^◡𝐺))
75		coundir 6198	. . 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 2793	. 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 14677	. . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐺:𝐷⟶𝐷) → (𝐺 ∘ 𝑊) ∈ Word 𝐷)
78	55, 66, 77	syl2anc 584	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ 𝑊) ∈ Word 𝐷)
79		f1of1 6780	. . . . . 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 6747	. . . . 5 ⊢ ((𝐺:𝐷–1-1→𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷) → (𝐺 ∘ 𝑊):dom 𝑊–1-1→𝐷)
82	80, 17, 81	syl2anc 584	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ 𝑊):dom 𝑊–1-1→𝐷)
83	66	fdmd 6676	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom 𝐺 = 𝐷)
84	20, 83	sseqtrrd 3983	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ran 𝑊 ⊆ dom 𝐺)
85		dmcosseq 5926	. . . . . 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 6731	. . . . 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 256	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ 𝑊):dom (𝐺 ∘ 𝑊)–1-1→𝐷)
90	6, 44, 78, 89	tocycfv 31782	. 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 2787	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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3405 ∖ cdif 3905 ∪ cun 3906 ⊆ wss 3908 I cid 5528 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 ∘ ccom 5635 ⟶wf 6489 –1-1→wf1 6490 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7351 1c1 11010 ℤcz 12457 Word cword 14355 cyclShift ccsh 14633 Basecbs 17042 +_gcplusg 17092 inv_gcminusg 18708 -_gcsg 18709 SymGrpcsymg 19106 toCycctocyc 31779

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 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 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-fz 13379 df-fzo 13522 df-fl 13651 df-mod 13729 df-hash 14184 df-word 14356 df-concat 14412 df-substr 14486 df-pfx 14516 df-csh 14634 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-tset 17111 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-submnd 18561 df-efmnd 18638 df-grp 18710 df-minusg 18711 df-sbg 18712 df-symg 19107 df-tocyc 31780

This theorem is referenced by: (None)

W3C validator