cycpmconjv - Mathbox for Thierry Arnoux

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Theorem cycpmconjv 33097

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 19272	. . . . . 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 33072	. . . . . . . . . . . 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 6666	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	5, 9	eleqtrd 2830	. . . . . . . . 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 5850	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
13		eqidd 2730	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
14	11, 12, 13	f1eq123d 6760	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
15	14	elrab 3650	. . . . . . . . 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 6724	. . . . . . 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 6664	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ran 𝑊 ⊆ 𝐷)
21	4, 20	cycpmconjvlem 33096	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) = ( I ↾ (𝐷 ∖ ran (𝐺 ↾ ran 𝑊))))
22		rnco 6205	. . . . . 6 ⊢ ran (𝐺 ∘ 𝑊) = ran (𝐺 ↾ ran 𝑊)
23	22	difeq2i 4076	. . . . 5 ⊢ (𝐷 ∖ ran (𝐺 ∘ 𝑊)) = (𝐷 ∖ ran (𝐺 ↾ ran 𝑊))
24	23	reseq2i 5931	. . . 4 ⊢ ( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))) = ( I ↾ (𝐷 ∖ ran (𝐺 ↾ ran 𝑊)))
25	21, 24	eqtr4di 2782	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∘ ^◡𝐺) = ( I ↾ (𝐷 ∖ ran (𝐺 ∘ 𝑊))))
26		coass 6218	. . . . 5 ⊢ ((((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊) ∘ ^◡𝐺) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ (^◡𝑊 ∘ ^◡𝐺))
27		cnvco 5832	. . . . . 6 ⊢ ^◡(𝐺 ∘ 𝑊) = (^◡𝑊 ∘ ^◡𝐺)
28	27	coeq2i 5807	. . . . 5 ⊢ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡(𝐺 ∘ 𝑊)) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ (^◡𝑊 ∘ ^◡𝐺))
29	26, 28	eqtr4i 2755	. . . 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 7023	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘𝑊) ∈ 𝐵)
34		cycpmconjv.p	. . . . . . . 8 ⊢ + = (+_g‘𝑆)
35	1, 2, 34	symgcl 19282	. . . . . . 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 2729	. . . . . . 7 ⊢ (inv_g‘𝑆) = (inv_g‘𝑆)
38		cycpmconjv.l	. . . . . . 7 ⊢ − = (-_g‘𝑆)
39	2, 34, 37, 38	grpsubval 18882	. . . . . 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 19299	. . . . . . 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 7369	. . . . 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 6780	. . . . . . . 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 19271	. . . . . . . 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 19281	. . . . . 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 2768	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = ((𝐺 + (𝑀‘𝑊)) ∘ ^◡𝐺))
53	1, 2, 34	symgov 19281	. . . . . . . 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 33064	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝑀‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)))
57	56	coeq2d 5809	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (𝑀‘𝑊)) = (𝐺 ∘ (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
58		coundi 6200	. . . . . . . 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 2768	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = ((𝐺 ∘ ( I ↾ (𝐷 ∖ ran 𝑊))) ∪ (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))))
61		coires1 6217	. . . . . . . 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 6218	. . . . . . . 8 ⊢ ((𝐺 ∘ (𝑊 cyclShift 1)) ∘ ^◡𝑊) = (𝐺 ∘ ((𝑊 cyclShift 1) ∘ ^◡𝑊))
64		1zzd 12524	. . . . . . . . . 10 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → 1 ∈ ℤ)
65		f1of 6768	. . . . . . . . . . 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 14761	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ ∧ 𝐺:𝐷⟶𝐷) → (𝐺 ∘ (𝑊 cyclShift 1)) = ((𝐺 ∘ 𝑊) cyclShift 1))
68	55, 64, 66, 67	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 ∘ (𝑊 cyclShift 1)) = ((𝐺 ∘ 𝑊) cyclShift 1))
69	68	coeq1d 5808	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 ∘ (𝑊 cyclShift 1)) ∘ ^◡𝑊) = (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊))
70	63, 69	eqtr3id 2778	. . . . . . 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 2764	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → (𝐺 + (𝑀‘𝑊)) = ((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)))
73	72	coeq1d 5808	. . . 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 2764	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (((𝐺 ↾ (𝐷 ∖ ran 𝑊)) ∪ (((𝐺 ∘ 𝑊) cyclShift 1) ∘ ^◡𝑊)) ∘ ^◡𝐺))
75		coundir 6201	. . 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 2780	. 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 14756	. . . 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 6767	. . . . . 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 6735	. . . . 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 6666	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → dom 𝐺 = 𝐷)
84	20, 83	sseqtrrd 3975	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ran 𝑊 ⊆ dom 𝐺)
85		dmcosseq 5923	. . . . . 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 6720	. . . . 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 33064	. 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 2774	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 1540 ∈ wcel 2109 {crab 3396 ∖ cdif 3902 ∪ cun 3903 ⊆ wss 3905 I cid 5517 ^◡ccnv 5622 dom cdm 5623 ran crn 5624 ↾ cres 5625 ∘ ccom 5627 ⟶wf 6482 –1-1→wf1 6483 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7353 1c1 11029 ℤcz 12489 Word cword 14438 cyclShift ccsh 14712 Basecbs 17138 +_gcplusg 17179 inv_gcminusg 18831 -_gcsg 18832 SymGrpcsymg 19266 toCycctocyc 33061

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-uz 12754 df-rp 12912 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-hash 14256 df-word 14439 df-concat 14496 df-substr 14566 df-pfx 14596 df-csh 14713 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-tset 17198 df-0g 17363 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-efmnd 18761 df-grp 18833 df-minusg 18834 df-sbg 18835 df-symg 19267 df-tocyc 33062

This theorem is referenced by: (None)

W3C validator