Theorem cyc3co2 33580

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 33575	. . . 4 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
11		eqid 2760	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
12	2, 11	symgbasf 19503	. . . 4 ⊢ ((𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷⟶𝐷)
13	10, 12	syl 18	. . 3 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷⟶𝐷)
14	13	ffnd 6703	. 2 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) Fn 𝐷)
15	2	symggrp 19527	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
16	3, 15	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Grp)
17	9	necomd 3010	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐾)
18	1, 3, 4, 6, 17, 2	cycpm2cl 33560	. . . . 5 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆))
19	1, 3, 4, 5, 7, 2	cycpm2cl 33560	. . . . 5 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
20		cyc3co2.t	. . . . . 6 ⊢ · = (+g‘𝑆)
21	11, 20	grpcl 19065	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
22	16, 18, 19, 21	syl3anc 1398	. . . 4 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
23	2, 11	symgbasf 19503	. . . 4 ⊢ (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)):𝐷⟶𝐷)
24	22, 23	syl 18	. . 3 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)):𝐷⟶𝐷)
25	24	ffnd 6703	. 2 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
26	1, 2, 3, 4, 5, 6, 7, 8, 9	cyc3fv1 33577	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
27	26	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
28		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 = 𝐼)
29	28	fveq2d 6882	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼))
30	2, 11, 20	symgov 19511	. . . . . . . . . . 11 ⊢ (((𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
31	18, 19, 30	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
32	31	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
33	32	fveq1d 6880	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
34	2, 11	symgbasf 19503	. . . . . . . . . . . 12 ⊢ ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
35	19, 34	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
36	35	ffund 6707	. . . . . . . . . 10 ⊢ (𝜑 → Fun (𝐶‘⟨“𝐼𝐽”⟩))
37	4	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝐼 ∈ 𝐷)
38	34	fdmd 6713	. . . . . . . . . . . . 13 ⊢ ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
39	19, 38	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
40	39	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
41	37, 28, 40	3eltr4d 2875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
42		fvco 6976	. . . . . . . . . 10 ⊢ ((Fun (𝐶‘⟨“𝐼𝐽”⟩) ∧ 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩)) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
43	36, 41, 42	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
44	28	fveq2d 6882	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼))
45	1, 3, 4, 5, 7, 2	cyc2fv1 33561	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
46	45	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
47	44, 46	eqtrd 2795	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐽)
48	47	fveq2d 6882	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽))
49	8	necomd 3010	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ≠ 𝐽)
50	7	necomd 3010	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ≠ 𝐼)
51	1, 2, 3, 4, 6, 5, 17, 49, 50	cyc2fvx 33574	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
52	51	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
53	43, 48, 52	3eqtrd 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
54	33, 53	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
55	27, 29, 54	3eqtr4d 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
56	55	adantlr 728	. . . . 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 33578	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
58	57	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
59		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝑥 = 𝐽)
60	59	fveq2d 6882	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽))
61	31	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
62	61	fveq1d 6880	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
63	5	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝐽 ∈ 𝐷)
64	39	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
65	63, 59, 64	3eltr4d 2875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
66	36, 65, 42	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
67	59	fveq2d 6882	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽))
68	1, 3, 4, 5, 7, 2	cyc2fv2 33562	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
69	68	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
70	67, 69	eqtrd 2795	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐼)
71	70	fveq2d 6882	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼))
72	1, 3, 4, 6, 17, 2	cyc2fv1 33561	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
73	72	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
74	66, 71, 73	3eqtrd 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
75	62, 74	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
76	58, 60, 75	3eqtr4d 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
77	76	adantlr 728	. . . . 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 33579	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
79	78	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
80		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝑥 = 𝐾)
81	80	fveq2d 6882	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾))
82	31	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
83	82	fveq1d 6880	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
84	6	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝐾 ∈ 𝐷)
85	39	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
86	84, 80, 85	3eltr4d 2875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
87	36, 86, 42	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
88	80	fveq2d 6882	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾))
89	1, 2, 3, 4, 5, 6, 7, 8, 9	cyc2fvx 33574	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
90	89	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
91	88, 90	eqtrd 2795	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐾)
92	91	fveq2d 6882	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾))
93	1, 3, 4, 6, 17, 2	cyc2fv2 33562	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
94	93	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
95	87, 92, 94	3eqtrd 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
96	83, 95	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
97	79, 81, 96	3eqtr4d 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
98	97	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
99		eltpi 4649	. . . . . 6 ⊢ (𝑥 ∈ {𝐼, 𝐽, 𝐾} → (𝑥 = 𝐼 ∨ 𝑥 = 𝐽 ∨ 𝑥 = 𝐾))
100	99	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → (𝑥 = 𝐼 ∨ 𝑥 = 𝐽 ∨ 𝑥 = 𝐾))
101	56, 77, 98, 100	mpjao3dan 1459	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
102	101	adantlr 728	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
103	35	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
104	103	ffund 6707	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → Fun (𝐶‘⟨“𝐼𝐽”⟩))
105		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾}))
106	105	eldifad 3911	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ 𝐷)
107	39	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
108	106, 107	eleqtrrd 2863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
109	104, 108, 42	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
110	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝐷 ∈ 𝑉)
111	4, 5	s2cld 14942	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
112	111	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
113	4, 5, 7	s2f1 33389	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
114	113	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
115		tpid1g 4730	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝐷 → 𝐼 ∈ {𝐼, 𝐽, 𝐾})
116	4, 115	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ {𝐼, 𝐽, 𝐾})
117		tpid2g 4732	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ 𝐷 → 𝐽 ∈ {𝐼, 𝐽, 𝐾})
118	5, 117	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ {𝐼, 𝐽, 𝐾})
119	116, 118	prssd 4783	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ {𝐼, 𝐽, 𝐾})
120	4, 5	s2rn 15036	. . . . . . . . . . . 12 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
121	120	eqcomd 2766	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} = ran ⟨“𝐼𝐽”⟩)
122	4, 5, 6	s3rn 15037	. . . . . . . . . . . 12 ⊢ (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
123	122	eqcomd 2766	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
124	119, 121, 123	3sstr3d 3985	. . . . . . . . . 10 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
125	124	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
126	105	eldifbd 3912	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ {𝐼, 𝐽, 𝐾})
127	122	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
128	126, 127	neleqtrrd 2883	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽𝐾”⟩)
129	125, 128	ssneldd 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽”⟩)
130	1, 110, 112, 114, 106, 129	cycpmfv3 33555	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝑥)
131	130	fveq2d 6882	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥))
132	4, 6	s2cld 14942	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
133	132	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
134	4, 6, 17	s2f1 33389	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1→𝐷)
135	134	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1→𝐷)
136		tpid3g 4733	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ {𝐼, 𝐽, 𝐾})
137	6, 136	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ {𝐼, 𝐽, 𝐾})
138	116, 137	prssd 4783	. . . . . . . . . 10 ⊢ (𝜑 → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
139	138	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
140	4, 6	s2rn 15036	. . . . . . . . . . 11 ⊢ (𝜑 → ran ⟨“𝐼𝐾”⟩ = {𝐼, 𝐾})
141	140	eqcomd 2766	. . . . . . . . . 10 ⊢ (𝜑 → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
142	141	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
143	123	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
144	139, 142, 143	3sstr3d 3985	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐾”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
145	144, 128	ssneldd 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐾”⟩)
146	1, 110, 133, 135, 106, 145	cycpmfv3 33555	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥) = 𝑥)
147	109, 131, 146	3eqtrd 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝑥)
148	31	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
149	148	fveq1d 6880	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
150	4, 5, 6	s3cld 14943	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
151	150	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
152	4, 5, 6, 7, 8, 9	s3f1 33390	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)
153	152	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)
154	1, 110, 151, 153, 106, 128	cycpmfv3 33555	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = 𝑥)
155	147, 149, 154	3eqtr4rd 2806	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
156	155	adantlr 728	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
157		tpssi 4798	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷 ∧ 𝐾 ∈ 𝐷) → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
158	4, 5, 6, 157	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
159		undif 4438	. . . . . . 7 ⊢ ({𝐼, 𝐽, 𝐾} ⊆ 𝐷 ↔ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
160	158, 159	sylib 221	. . . . . 6 ⊢ (𝜑 → ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
161	160	eleq2d 2846	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ 𝑥 ∈ 𝐷))
162	161	biimpar 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
163		elun 4100	. . . 4 ⊢ (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
164	162, 163	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
165	102, 156, 164	mpjaodan 973	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
166	14, 25, 165	eqfnfvd 7025	1 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  {cpr 4586  {ctp 4588  dom cdm 5655  ran crn 5656   ∘ ccom 5659  Fun wfun 6527  ⟶wf 6529  –1-1→wf1 6530  ‘cfv 6533  (class class class)co 7413  Word cword 14578  ⟨“cs2 14912  ⟨“cs3 14913  Basecbs 17301  +_gcplusg 17342  Grpcgrp 19057  SymGrpcsymg 19496  toCycctocyc 33546

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-uz 12888  df-rp 13043  df-fz 13562  df-fzo 13710  df-fl 13853  df-mod 13931  df-hash 14395  df-word 14579  df-concat 14636  df-s1 14663  df-substr 14709  df-pfx 14741  df-csh 14860  df-s2 14919  df-s3 14920  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-tset 17361  df-0g 17526  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-efmnd 18978  df-grp 19060  df-symg 19497  df-tocyc 33547

This theorem is used by:  cyc3evpm  33590  cyc3genpmlem  33591