Theorem cyc3co2 33097

Description: Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)

Hypotheses

Ref	Expression
cycpm3.c	⊢ 𝐶 = (toCyc‘𝐷)
cycpm3.s	⊢ 𝑆 = (SymGrp‘𝐷)
cycpm3.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
cycpm3.i	⊢ (𝜑 → 𝐼 ∈ 𝐷)
cycpm3.j	⊢ (𝜑 → 𝐽 ∈ 𝐷)
cycpm3.k	⊢ (𝜑 → 𝐾 ∈ 𝐷)
cycpm3.1	⊢ (𝜑 → 𝐼 ≠ 𝐽)
cycpm3.2	⊢ (𝜑 → 𝐽 ≠ 𝐾)
cycpm3.3	⊢ (𝜑 → 𝐾 ≠ 𝐼)
cyc3co2.t	⊢ · = (+g‘𝑆)

Assertion

Ref	Expression
cyc3co2	⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))

Proof of Theorem cyc3co2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpm3.c	. . . . 5 ⊢ 𝐶 = (toCyc‘𝐷)
2		cycpm3.s	. . . . 5 ⊢ 𝑆 = (SymGrp‘𝐷)
3		cycpm3.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
4		cycpm3.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
5		cycpm3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
6		cycpm3.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
7		cycpm3.1	. . . . 5 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
8		cycpm3.2	. . . . 5 ⊢ (𝜑 → 𝐽 ≠ 𝐾)
9		cycpm3.3	. . . . 5 ⊢ (𝜑 → 𝐾 ≠ 𝐼)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	cycpm3cl 33092	. . . 4 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
11		eqid 2729	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
12	2, 11	symgbasf 19306	. . . 4 ⊢ ((𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷⟶𝐷)
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷⟶𝐷)
14	13	ffnd 6689	. 2 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) Fn 𝐷)
15	2	symggrp 19330	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Grp)
17	9	necomd 2980	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐾)
18	1, 3, 4, 6, 17, 2	cycpm2cl 33077	. . . . 5 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆))
19	1, 3, 4, 5, 7, 2	cycpm2cl 33077	. . . . 5 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
20		cyc3co2.t	. . . . . 6 ⊢ · = (+g‘𝑆)
21	11, 20	grpcl 18873	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
22	16, 18, 19, 21	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
23	2, 11	symgbasf 19306	. . . 4 ⊢ (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)):𝐷⟶𝐷)
24	22, 23	syl 17	. . 3 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)):𝐷⟶𝐷)
25	24	ffnd 6689	. 2 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
26	1, 2, 3, 4, 5, 6, 7, 8, 9	cyc3fv1 33094	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
27	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
28		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 = 𝐼)
29	28	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼))
30	2, 11, 20	symgov 19314	. . . . . . . . . . 11 ⊢ (((𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
31	18, 19, 30	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
32	31	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
33	32	fveq1d 6860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
34	2, 11	symgbasf 19306	. . . . . . . . . . . 12 ⊢ ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
35	19, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
36	35	ffund 6692	. . . . . . . . . 10 ⊢ (𝜑 → Fun (𝐶‘⟨“𝐼𝐽”⟩))
37	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝐼 ∈ 𝐷)
38	34	fdmd 6698	. . . . . . . . . . . . 13 ⊢ ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
39	19, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
41	37, 28, 40	3eltr4d 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
42		fvco 6959	. . . . . . . . . 10 ⊢ ((Fun (𝐶‘⟨“𝐼𝐽”⟩) ∧ 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩)) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
43	36, 41, 42	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
44	28	fveq2d 6862	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼))
45	1, 3, 4, 5, 7, 2	cyc2fv1 33078	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
46	45	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
47	44, 46	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐽)
48	47	fveq2d 6862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽))
49	8	necomd 2980	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ≠ 𝐽)
50	7	necomd 2980	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ≠ 𝐼)
51	1, 2, 3, 4, 6, 5, 17, 49, 50	cyc2fvx 33091	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
52	51	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
53	43, 48, 52	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
54	33, 53	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
55	27, 29, 54	3eqtr4d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
56	55	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
57	1, 2, 3, 4, 5, 6, 7, 8, 9	cyc3fv2 33095	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
58	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
59		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝑥 = 𝐽)
60	59	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽))
61	31	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
62	61	fveq1d 6860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
63	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝐽 ∈ 𝐷)
64	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
65	63, 59, 64	3eltr4d 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
66	36, 65, 42	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
67	59	fveq2d 6862	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽))
68	1, 3, 4, 5, 7, 2	cyc2fv2 33079	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
69	68	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
70	67, 69	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐼)
71	70	fveq2d 6862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼))
72	1, 3, 4, 6, 17, 2	cyc2fv1 33078	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
73	72	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
74	66, 71, 73	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
75	62, 74	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
76	58, 60, 75	3eqtr4d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
77	76	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
78	1, 2, 3, 4, 5, 6, 7, 8, 9	cyc3fv3 33096	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
79	78	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
80		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝑥 = 𝐾)
81	80	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾))
82	31	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
83	82	fveq1d 6860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
84	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝐾 ∈ 𝐷)
85	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
86	84, 80, 85	3eltr4d 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
87	36, 86, 42	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
88	80	fveq2d 6862	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾))
89	1, 2, 3, 4, 5, 6, 7, 8, 9	cyc2fvx 33091	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
90	89	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
91	88, 90	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐾)
92	91	fveq2d 6862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾))
93	1, 3, 4, 6, 17, 2	cyc2fv2 33079	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
94	93	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
95	87, 92, 94	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
96	83, 95	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
97	79, 81, 96	3eqtr4d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
98	97	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
99		eltpi 4652	. . . . . 6 ⊢ (𝑥 ∈ {𝐼, 𝐽, 𝐾} → (𝑥 = 𝐼 ∨ 𝑥 = 𝐽 ∨ 𝑥 = 𝐾))
100	99	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → (𝑥 = 𝐼 ∨ 𝑥 = 𝐽 ∨ 𝑥 = 𝐾))
101	56, 77, 98, 100	mpjao3dan 1434	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
102	101	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
103	35	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
104	103	ffund 6692	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → Fun (𝐶‘⟨“𝐼𝐽”⟩))
105		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾}))
106	105	eldifad 3926	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ 𝐷)
107	39	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
108	106, 107	eleqtrrd 2831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
109	104, 108, 42	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
110	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝐷 ∈ 𝑉)
111	4, 5	s2cld 14837	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
112	111	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
113	4, 5, 7	s2f1 32866	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
114	113	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
115		tpid1g 4733	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝐷 → 𝐼 ∈ {𝐼, 𝐽, 𝐾})
116	4, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ {𝐼, 𝐽, 𝐾})
117		tpid2g 4735	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ 𝐷 → 𝐽 ∈ {𝐼, 𝐽, 𝐾})
118	5, 117	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ {𝐼, 𝐽, 𝐾})
119	116, 118	prssd 4786	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ {𝐼, 𝐽, 𝐾})
120	4, 5	s2rn 14929	. . . . . . . . . . . 12 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
121	120	eqcomd 2735	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} = ran ⟨“𝐼𝐽”⟩)
122	4, 5, 6	s3rn 14930	. . . . . . . . . . . 12 ⊢ (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
123	122	eqcomd 2735	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
124	119, 121, 123	3sstr3d 4001	. . . . . . . . . 10 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
125	124	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
126	105	eldifbd 3927	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ {𝐼, 𝐽, 𝐾})
127	122	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
128	126, 127	neleqtrrd 2851	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽𝐾”⟩)
129	125, 128	ssneldd 3949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽”⟩)
130	1, 110, 112, 114, 106, 129	cycpmfv3 33072	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝑥)
131	130	fveq2d 6862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥))
132	4, 6	s2cld 14837	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
133	132	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
134	4, 6, 17	s2f1 32866	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1→𝐷)
135	134	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1→𝐷)
136		tpid3g 4736	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ {𝐼, 𝐽, 𝐾})
137	6, 136	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ {𝐼, 𝐽, 𝐾})
138	116, 137	prssd 4786	. . . . . . . . . 10 ⊢ (𝜑 → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
139	138	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
140	4, 6	s2rn 14929	. . . . . . . . . . 11 ⊢ (𝜑 → ran ⟨“𝐼𝐾”⟩ = {𝐼, 𝐾})
141	140	eqcomd 2735	. . . . . . . . . 10 ⊢ (𝜑 → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
142	141	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
143	123	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
144	139, 142, 143	3sstr3d 4001	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐾”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
145	144, 128	ssneldd 3949	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐾”⟩)
146	1, 110, 133, 135, 106, 145	cycpmfv3 33072	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥) = 𝑥)
147	109, 131, 146	3eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝑥)
148	31	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
149	148	fveq1d 6860	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
150	4, 5, 6	s3cld 14838	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
151	150	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
152	4, 5, 6, 7, 8, 9	s3f1 32868	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)
153	152	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)
154	1, 110, 151, 153, 106, 128	cycpmfv3 33072	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = 𝑥)
155	147, 149, 154	3eqtr4rd 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
156	155	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
157		tpssi 4802	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷 ∧ 𝐾 ∈ 𝐷) → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
158	4, 5, 6, 157	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
159		undif 4445	. . . . . . 7 ⊢ ({𝐼, 𝐽, 𝐾} ⊆ 𝐷 ↔ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
160	158, 159	sylib 218	. . . . . 6 ⊢ (𝜑 → ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
161	160	eleq2d 2814	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ 𝑥 ∈ 𝐷))
162	161	biimpar 477	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
163		elun 4116	. . . 4 ⊢ (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
164	162, 163	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
165	102, 156, 164	mpjaodan 960	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
166	14, 25, 165	eqfnfvd 7006	1 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3911   ∪ cun 3912   ⊆ wss 3914  {cpr 4591  {ctp 4593  dom cdm 5638  ran crn 5639   ∘ ccom 5642  Fun wfun 6505  ⟶wf 6507  –1-1→wf1 6508  ‘cfv 6511  (class class class)co 7387  Word cword 14478  ⟨“cs2 14807  ⟨“cs3 14808  Basecbs 17179  +_gcplusg 17220  Grpcgrp 18865  SymGrpcsymg 19299  toCycctocyc 33063

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-substr 14606  df-pfx 14636  df-csh 14754  df-s2 14814  df-s3 14815  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-tset 17239  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-efmnd 18796  df-grp 18868  df-symg 19300  df-tocyc 33064

This theorem is referenced by:  cyc3evpm  33107  cyc3genpmlem  33108