cyc3co2 - Mathbox for Thierry Arnoux

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Theorem cyc3co2 32557

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 32552	. . . 4 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
11		eqid 2732	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
12	2, 11	symgbasf 19284	. . . 4 ⊢ ((𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷⟶𝐷)
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷⟶𝐷)
14	13	ffnd 6718	. 2 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) Fn 𝐷)
15	2	symggrp 19309	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Grp)
17	9	necomd 2996	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐾)
18	1, 3, 4, 6, 17, 2	cycpm2cl 32537	. . . . 5 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆))
19	1, 3, 4, 5, 7, 2	cycpm2cl 32537	. . . . 5 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
20		cyc3co2.t	. . . . . 6 ⊢ · = (+_g‘𝑆)
21	11, 20	grpcl 18863	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
22	16, 18, 19, 21	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
23	2, 11	symgbasf 19284	. . . 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 6718	. 2 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
26	1, 2, 3, 4, 5, 6, 7, 8, 9	cyc3fv1 32554	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
27	26	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
28		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 = 𝐼)
29	28	fveq2d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼))
30	2, 11, 20	symgov 19292	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
33	32	fveq1d 6893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
34	2, 11	symgbasf 19284	. . . . . . . . . . . 12 ⊢ ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
35	19, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
36	35	ffund 6721	. . . . . . . . . 10 ⊢ (𝜑 → Fun (𝐶‘⟨“𝐼𝐽”⟩))
37	4	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝐼 ∈ 𝐷)
38	34	fdmd 6728	. . . . . . . . . . . . 13 ⊢ ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
39	19, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
40	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
41	37, 28, 40	3eltr4d 2848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
42		fvco 6989	. . . . . . . . . 10 ⊢ ((Fun (𝐶‘⟨“𝐼𝐽”⟩) ∧ 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩)) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
43	36, 41, 42	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
44	28	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼))
45	1, 3, 4, 5, 7, 2	cyc2fv1 32538	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
46	45	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
47	44, 46	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐽)
48	47	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽))
49	8	necomd 2996	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ≠ 𝐽)
50	7	necomd 2996	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ≠ 𝐼)
51	1, 2, 3, 4, 6, 5, 17, 49, 50	cyc2fvx 32551	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
52	51	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
53	43, 48, 52	3eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
54	33, 53	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
55	27, 29, 54	3eqtr4d 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
56	55	adantlr 713	. . . . 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 32555	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
58	57	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
59		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝑥 = 𝐽)
60	59	fveq2d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽))
61	31	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
62	61	fveq1d 6893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
63	5	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝐽 ∈ 𝐷)
64	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
65	63, 59, 64	3eltr4d 2848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
66	36, 65, 42	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
67	59	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽))
68	1, 3, 4, 5, 7, 2	cyc2fv2 32539	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
69	68	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
70	67, 69	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐼)
71	70	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼))
72	1, 3, 4, 6, 17, 2	cyc2fv1 32538	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
73	72	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
74	66, 71, 73	3eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
75	62, 74	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
76	58, 60, 75	3eqtr4d 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
77	76	adantlr 713	. . . . 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 32556	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
79	78	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
80		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝑥 = 𝐾)
81	80	fveq2d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾))
82	31	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
83	82	fveq1d 6893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
84	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝐾 ∈ 𝐷)
85	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
86	84, 80, 85	3eltr4d 2848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
87	36, 86, 42	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
88	80	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾))
89	1, 2, 3, 4, 5, 6, 7, 8, 9	cyc2fvx 32551	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
90	89	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
91	88, 90	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐾)
92	91	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾))
93	1, 3, 4, 6, 17, 2	cyc2fv2 32539	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
94	93	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
95	87, 92, 94	3eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
96	83, 95	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
97	79, 81, 96	3eqtr4d 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
98	97	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
99		eltpi 4691	. . . . . 6 ⊢ (𝑥 ∈ {𝐼, 𝐽, 𝐾} → (𝑥 = 𝐼 ∨ 𝑥 = 𝐽 ∨ 𝑥 = 𝐾))
100	99	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → (𝑥 = 𝐼 ∨ 𝑥 = 𝐽 ∨ 𝑥 = 𝐾))
101	56, 77, 98, 100	mpjao3dan 1431	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
102	101	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
103	35	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
104	103	ffund 6721	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → Fun (𝐶‘⟨“𝐼𝐽”⟩))
105		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾}))
106	105	eldifad 3960	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ 𝐷)
107	39	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
108	106, 107	eleqtrrd 2836	. . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝐷 ∈ 𝑉)
111	4, 5	s2cld 14826	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
112	111	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
113	4, 5, 7	s2f1 32366	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
114	113	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
115		tpid1g 4773	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝐷 → 𝐼 ∈ {𝐼, 𝐽, 𝐾})
116	4, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ {𝐼, 𝐽, 𝐾})
117		tpid2g 4775	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ 𝐷 → 𝐽 ∈ {𝐼, 𝐽, 𝐾})
118	5, 117	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ {𝐼, 𝐽, 𝐾})
119	116, 118	prssd 4825	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ {𝐼, 𝐽, 𝐾})
120	4, 5	s2rn 32365	. . . . . . . . . . . 12 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
121	120	eqcomd 2738	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽} = ran ⟨“𝐼𝐽”⟩)
122	4, 5, 6	s3rn 32367	. . . . . . . . . . . 12 ⊢ (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
123	122	eqcomd 2738	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
124	119, 121, 123	3sstr3d 4028	. . . . . . . . . 10 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
125	124	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
126	105	eldifbd 3961	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ {𝐼, 𝐽, 𝐾})
127	122	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
128	126, 127	neleqtrrd 2856	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽𝐾”⟩)
129	125, 128	ssneldd 3985	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽”⟩)
130	1, 110, 112, 114, 106, 129	cycpmfv3 32532	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝑥)
131	130	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥))
132	4, 6	s2cld 14826	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
133	132	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
134	4, 6, 17	s2f1 32366	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1→𝐷)
135	134	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1→𝐷)
136		tpid3g 4776	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ {𝐼, 𝐽, 𝐾})
137	6, 136	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ {𝐼, 𝐽, 𝐾})
138	116, 137	prssd 4825	. . . . . . . . . 10 ⊢ (𝜑 → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
139	138	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
140	4, 6	s2rn 32365	. . . . . . . . . . 11 ⊢ (𝜑 → ran ⟨“𝐼𝐾”⟩ = {𝐼, 𝐾})
141	140	eqcomd 2738	. . . . . . . . . 10 ⊢ (𝜑 → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
142	141	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
143	123	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
144	139, 142, 143	3sstr3d 4028	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐾”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
145	144, 128	ssneldd 3985	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐾”⟩)
146	1, 110, 133, 135, 106, 145	cycpmfv3 32532	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥) = 𝑥)
147	109, 131, 146	3eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝑥)
148	31	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
149	148	fveq1d 6893	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
150	4, 5, 6	s3cld 14827	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
151	150	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
152	4, 5, 6, 7, 8, 9	s3f1 32368	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)
153	152	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)
154	1, 110, 151, 153, 106, 128	cycpmfv3 32532	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = 𝑥)
155	147, 149, 154	3eqtr4rd 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
156	155	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
157		tpssi 4839	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷 ∧ 𝐾 ∈ 𝐷) → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
158	4, 5, 6, 157	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
159		undif 4481	. . . . . . 7 ⊢ ({𝐼, 𝐽, 𝐾} ⊆ 𝐷 ↔ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
160	158, 159	sylib 217	. . . . . 6 ⊢ (𝜑 → ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
161	160	eleq2d 2819	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ 𝑥 ∈ 𝐷))
162	161	biimpar 478	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
163		elun 4148	. . . 4 ⊢ (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
164	162, 163	sylib 217	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
165	102, 156, 164	mpjaodan 957	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
166	14, 25, 165	eqfnfvd 7035	1 ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 {cpr 4630 {ctp 4632 dom cdm 5676 ran crn 5677 ∘ ccom 5680 Fun wfun 6537 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7411 Word cword 14468 ⟨“cs2 14796 ⟨“cs3 14797 Basecbs 17148 +_gcplusg 17201 Grpcgrp 18855 SymGrpcsymg 19275 toCycctocyc 32523

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-substr 14595 df-pfx 14625 df-csh 14743 df-s2 14803 df-s3 14804 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-tset 17220 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-efmnd 18786 df-grp 18858 df-symg 19276 df-tocyc 32524

This theorem is referenced by: cyc3evpm 32567 cyc3genpmlem 32568

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