cycpmconjv - Mathbox for Thierry Arnoux

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Theorem cycpmconjv 33163

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 19393	. . . . . 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 33138	. . . . . . . . . . . 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 6745	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	5, 9	eleqtrd 2842	. . . . . . . . 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 5913	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
13		eqidd 2737	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
14	11, 12, 13	f1eq123d 6839	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
15	14	elrab 3691	. . . . . . . . 9 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
16	10, 15	sylib 218	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
17	16	simprd 495	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊:dom 𝑊–1-1→𝐷)
18		f1f 6803	. . . . . . 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 6743	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ran 𝑊 ⊆ 𝐷)
21	4, 20	cycpmconjvlem 33162	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) = ( I ↾ (𝐷 ∖ ran (𝐺 ↾ ran 𝑊))))
22		rnco 6271	. . . . . 6 ⊢ ran (𝐺 ∘ 𝑊) = ran (𝐺 ↾ ran 𝑊)
23	22	difeq2i 4122	. . . . 5 ⊢ (𝐷 ∖ ran (𝐺 ∘ 𝑊)) = (𝐷 ∖ ran (𝐺 ↾ ran 𝑊))
24	23	reseq2i 5993	. . . 4 ⊢ ( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))) = ( I ↾ (𝐷 ∖ ran (𝐺 ↾ ran 𝑊)))
25	21, 24	eqtr4di 2794	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) = ( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))))
26		coass 6284	. . . . 5 ⊢ ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ (^◡𝑊 ∘ ^◡𝐺))
27		cnvco 5895	. . . . . 6 ⊢ ^◡(𝐺 ∘ 𝑊) = (^◡𝑊 ∘ ^◡𝐺)
28	27	coeq2i 5870	. . . . 5 ⊢ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡(𝐺 ∘ 𝑊)) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ (^◡𝑊 ∘ ^◡𝐺))
29	26, 28	eqtr4i 2767	. . . 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 4168	. 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 7104	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘𝑊) ∈ 𝐵)
34		cycpmconjv.p	. . . . . . . 8 ⊢ + = (+_g‘𝑆)
35	1, 2, 34	symgcl 19403	. . . . . . 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 2736	. . . . . . 7 ⊢ (inv_g‘𝑆) = (inv_g‘𝑆)
38		cycpmconjv.l	. . . . . . 7 ⊢ − = (-_g‘𝑆)
39	2, 34, 37, 38	grpsubval 19004	. . . . . 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 19421	. . . . . . 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 7448	. . . . 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 6859	. . . . . . . 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 19392	. . . . . . . 8 ⊢ (𝐷 ∈ 𝑉 → (^◡𝐺 ∈ 𝐵 ↔ ^◡𝐺:𝐷–1-1-onto→𝐷))
48	47	biimpar 477	. . . . . . 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 19402	. . . . . 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 2780	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺))
53	1, 2, 34	symgov 19402	. . . . . . . 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 494	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 𝑊 ∈ Word 𝐷)
56	6, 44, 55, 17	tocycfv 33130	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)))
57	56	coeq2d 5872	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (𝑀‘𝑊)) = (𝐺 ∘ (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
58		coundi 6266	. . . . . . . 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 2780	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = ((𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) ∪ (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
61		coires1 6283	. . . . . . . 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 6284	. . . . . . . 8 ⊢ ((𝐺 ∘ (𝑊 cyclShift 1)) ∘ ^◡𝑊) = (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))
64		1zzd 12650	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 1 ∈ ℤ)
65		f1of 6847	. . . . . . . . . . 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 14876	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ ∧ 𝐺:𝐷⟶𝐷) → (𝐺 ∘ (𝑊 cyclShift 1)) = ((𝐺 ∘ 𝑊) cyclShift 1))
68	55, 64, 66, 67	syl3anc 1372	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (𝑊 cyclShift 1)) = ((𝐺 ∘ 𝑊) cyclShift 1))
69	68	coeq1d 5871	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ∘ (𝑊 cyclShift 1)) ∘ ^◡𝑊) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊))
70	63, 69	eqtr3id 2790	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊))
71	62, 70	uneq12d 4168	. . . . . 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 2776	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)))
73	72	coeq1d 5871	. . . 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 2776	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)) ∘ ^◡𝐺))
75		coundir 6267	. . 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 2792	. 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 14871	. . . 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 6846	. . . . . 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 6814	. . . . 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 6745	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom 𝐺 = 𝐷)
84	20, 83	sseqtrrd 4020	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ran 𝑊 ⊆ dom 𝐺)
85		dmcosseq 5986	. . . . . 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 6799	. . . . 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 257	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ 𝑊):dom (𝐺 ∘ 𝑊)–1-1→𝐷)
90	6, 44, 78, 89	tocycfv 33130	. 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 2786	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {crab 3435 ∖ cdif 3947 ∪ cun 3948 ⊆ wss 3950 I cid 5576 ^◡ccnv 5683 dom cdm 5684 ran crn 5685 ↾ cres 5686 ∘ ccom 5688 ⟶wf 6556 –1-1→wf1 6557 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 1c1 11157 ℤcz 12615 Word cword 14553 cyclShift ccsh 14827 Basecbs 17248 +_gcplusg 17298 inv_gcminusg 18953 -_gcsg 18954 SymGrpcsymg 19387 toCycctocyc 33127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-hash 14371 df-word 14554 df-concat 14610 df-substr 14680 df-pfx 14710 df-csh 14828 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-tset 17317 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-efmnd 18883 df-grp 18955 df-minusg 18956 df-sbg 18957 df-symg 19388 df-tocyc 33128

This theorem is referenced by: (None)

W3C validator