rhmpropd - Intuitionistic Logic Explorer

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Theorem rhmpropd 14261

Description: Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)

**Hypotheses**
Ref	Expression
rhmpropd.a	⊢ (𝜑 → 𝐵 = (Base‘𝐽))
rhmpropd.b	⊢ (𝜑 → 𝐶 = (Base‘𝐾))
rhmpropd.c	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
rhmpropd.d	⊢ (𝜑 → 𝐶 = (Base‘𝑀))
rhmpropd.e	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐽)𝑦) = (𝑥(+_g‘𝐿)𝑦))
rhmpropd.f	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝑀)𝑦))
rhmpropd.g	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐽)𝑦) = (𝑥(._r‘𝐿)𝑦))
rhmpropd.h	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝑀)𝑦))

**Assertion**
Ref	Expression
rhmpropd	⊢ (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀))

Distinct variable groups: 𝑥,𝑦,𝐽 𝑥,𝐾,𝑦 𝑥,𝐿,𝑦 𝑥,𝑀,𝑦 𝜑,𝑥,𝑦 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦

Proof of Theorem rhmpropd Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2229	. . . . . 6 ⊢ (mulGrp‘𝐽) = (mulGrp‘𝐽)
2		eqid 2229	. . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
3	1, 2	isrhm 14165	. . . . 5 ⊢ (𝑓 ∈ (𝐽 RingHom 𝐾) ↔ ((𝐽 ∈ Ring ∧ 𝐾 ∈ Ring) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝑓 ∈ ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)))))
4	3	simplbi 274	. . . 4 ⊢ (𝑓 ∈ (𝐽 RingHom 𝐾) → (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring))
5	4	a1i 9	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐽 RingHom 𝐾) → (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)))
6		eqid 2229	. . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
7		eqid 2229	. . . . . 6 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
8	6, 7	isrhm 14165	. . . . 5 ⊢ (𝑓 ∈ (𝐿 RingHom 𝑀) ↔ ((𝐿 ∈ Ring ∧ 𝑀 ∈ Ring) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ 𝑓 ∈ ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀)))))
9	8	simplbi 274	. . . 4 ⊢ (𝑓 ∈ (𝐿 RingHom 𝑀) → (𝐿 ∈ Ring ∧ 𝑀 ∈ Ring))
10		rhmpropd.a	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐽))
11		rhmpropd.c	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
12		rhmpropd.e	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐽)𝑦) = (𝑥(+_g‘𝐿)𝑦))
13		rhmpropd.g	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐽)𝑦) = (𝑥(._r‘𝐿)𝑦))
14	10, 11, 12, 13	ringpropd 14044	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ Ring ↔ 𝐿 ∈ Ring))
15		rhmpropd.b	. . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝐾))
16		rhmpropd.d	. . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝑀))
17		rhmpropd.f	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝑀)𝑦))
18		rhmpropd.h	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝑀)𝑦))
19	15, 16, 17, 18	ringpropd 14044	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝑀 ∈ Ring))
20	14, 19	anbi12d 473	. . . 4 ⊢ (𝜑 → ((𝐽 ∈ Ring ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ Ring ∧ 𝑀 ∈ Ring)))
21	9, 20	imbitrrid 156	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐿 RingHom 𝑀) → (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)))
22	20	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → ((𝐽 ∈ Ring ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ Ring ∧ 𝑀 ∈ Ring)))
23	10, 15, 11, 16, 12, 17	ghmpropd 13863	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
24	23	eleq2d 2299	. . . . . . . 8 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
25	24	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
26	10	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐽))
27		simprl 529	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐽 ∈ Ring)
28		eqid 2229	. . . . . . . . . . . 12 ⊢ (Base‘𝐽) = (Base‘𝐽)
29	1, 28	mgpbasg 13932	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Ring → (Base‘𝐽) = (Base‘(mulGrp‘𝐽)))
30	27, 29	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → (Base‘𝐽) = (Base‘(mulGrp‘𝐽)))
31	26, 30	eqtrd 2262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘(mulGrp‘𝐽)))
32	15	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐶 = (Base‘𝐾))
33		simprr 531	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐾 ∈ Ring)
34		eqid 2229	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
35	2, 34	mgpbasg 13932	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
36	33, 35	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
37	32, 36	eqtrd 2262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐶 = (Base‘(mulGrp‘𝐾)))
38	11	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐿))
39	20	simprbda 383	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐿 ∈ Ring)
40		eqid 2229	. . . . . . . . . . . 12 ⊢ (Base‘𝐿) = (Base‘𝐿)
41	6, 40	mgpbasg 13932	. . . . . . . . . . 11 ⊢ (𝐿 ∈ Ring → (Base‘𝐿) = (Base‘(mulGrp‘𝐿)))
42	39, 41	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → (Base‘𝐿) = (Base‘(mulGrp‘𝐿)))
43	38, 42	eqtrd 2262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘(mulGrp‘𝐿)))
44	16	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐶 = (Base‘𝑀))
45	20	simplbda 384	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝑀 ∈ Ring)
46		eqid 2229	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
47	7, 46	mgpbasg 13932	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Ring → (Base‘𝑀) = (Base‘(mulGrp‘𝑀)))
48	45, 47	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → (Base‘𝑀) = (Base‘(mulGrp‘𝑀)))
49	44, 48	eqtrd 2262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → 𝐶 = (Base‘(mulGrp‘𝑀)))
50	13	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐽)𝑦) = (𝑥(._r‘𝐿)𝑦))
51	27	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐽 ∈ Ring)
52		eqid 2229	. . . . . . . . . . . . 13 ⊢ (._r‘𝐽) = (._r‘𝐽)
53	1, 52	mgpplusgg 13930	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Ring → (._r‘𝐽) = (+_g‘(mulGrp‘𝐽)))
54	53	oveqd 6030	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Ring → (𝑥(._r‘𝐽)𝑦) = (𝑥(+_g‘(mulGrp‘𝐽))𝑦))
55	51, 54	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐽)𝑦) = (𝑥(+_g‘(mulGrp‘𝐽))𝑦))
56	39	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ Ring)
57		eqid 2229	. . . . . . . . . . . . 13 ⊢ (._r‘𝐿) = (._r‘𝐿)
58	6, 57	mgpplusgg 13930	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Ring → (._r‘𝐿) = (+_g‘(mulGrp‘𝐿)))
59	58	oveqd 6030	. . . . . . . . . . 11 ⊢ (𝐿 ∈ Ring → (𝑥(._r‘𝐿)𝑦) = (𝑥(+_g‘(mulGrp‘𝐿))𝑦))
60	56, 59	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐿)𝑦) = (𝑥(+_g‘(mulGrp‘𝐿))𝑦))
61	50, 55, 60	3eqtr3d 2270	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘(mulGrp‘𝐽))𝑦) = (𝑥(+_g‘(mulGrp‘𝐿))𝑦))
62	18	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝑀)𝑦))
63	33	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐾 ∈ Ring)
64		eqid 2229	. . . . . . . . . . . . 13 ⊢ (._r‘𝐾) = (._r‘𝐾)
65	2, 64	mgpplusgg 13930	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (._r‘𝐾) = (+_g‘(mulGrp‘𝐾)))
66	65	oveqd 6030	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → (𝑥(._r‘𝐾)𝑦) = (𝑥(+_g‘(mulGrp‘𝐾))𝑦))
67	63, 66	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐾)𝑦) = (𝑥(+_g‘(mulGrp‘𝐾))𝑦))
68	45	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑀 ∈ Ring)
69		eqid 2229	. . . . . . . . . . . . 13 ⊢ (._r‘𝑀) = (._r‘𝑀)
70	7, 69	mgpplusgg 13930	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ Ring → (._r‘𝑀) = (+_g‘(mulGrp‘𝑀)))
71	70	oveqd 6030	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Ring → (𝑥(._r‘𝑀)𝑦) = (𝑥(+_g‘(mulGrp‘𝑀))𝑦))
72	68, 71	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝑀)𝑦) = (𝑥(+_g‘(mulGrp‘𝑀))𝑦))
73	62, 67, 72	3eqtr3d 2270	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(+_g‘(mulGrp‘𝑀))𝑦))
74	31, 37, 43, 49, 61, 73	mhmpropd 13542	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)) = ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀)))
75	74	eleq2d 2299	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → (𝑓 ∈ ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)) ↔ 𝑓 ∈ ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀))))
76	25, 75	anbi12d 473	. . . . . 6 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝑓 ∈ ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾))) ↔ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ 𝑓 ∈ ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀)))))
77	22, 76	anbi12d 473	. . . . 5 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → (((𝐽 ∈ Ring ∧ 𝐾 ∈ Ring) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝑓 ∈ ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)))) ↔ ((𝐿 ∈ Ring ∧ 𝑀 ∈ Ring) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ 𝑓 ∈ ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀))))))
78	77, 3, 8	3bitr4g 223	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → (𝑓 ∈ (𝐽 RingHom 𝐾) ↔ 𝑓 ∈ (𝐿 RingHom 𝑀)))
79	78	ex 115	. . 3 ⊢ (𝜑 → ((𝐽 ∈ Ring ∧ 𝐾 ∈ Ring) → (𝑓 ∈ (𝐽 RingHom 𝐾) ↔ 𝑓 ∈ (𝐿 RingHom 𝑀))))
80	5, 21, 79	pm5.21ndd 710	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 RingHom 𝐾) ↔ 𝑓 ∈ (𝐿 RingHom 𝑀)))
81	80	eqrdv 2227	1 ⊢ (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1395 ∈ wcel 2200 ‘cfv 5324 (class class class)co 6013 Basecbs 13075 +_gcplusg 13153 ._rcmulr 13154 MndHom cmhm 13533 GrpHom cghm 13820 mulGrpcmgp 13926 Ringcrg 14002 RingHom crh 14157

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-addcom 8125 ax-addass 8127 ax-i2m1 8130 ax-0lt1 8131 ax-0id 8133 ax-rnegex 8134 ax-pre-ltirr 8137 ax-pre-ltadd 8141

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-map 6814 df-pnf 8209 df-mnf 8210 df-ltxr 8212 df-inn 9137 df-2 9195 df-3 9196 df-ndx 13078 df-slot 13079 df-base 13081 df-sets 13082 df-plusg 13166 df-mulr 13167 df-0g 13334 df-mgm 13432 df-sgrp 13478 df-mnd 13493 df-mhm 13535 df-grp 13579 df-ghm 13821 df-mgp 13927 df-ur 13966 df-ring 14004 df-rhm 14159

This theorem is referenced by: zrhpropd 14633

W3C validator