rhmpropd - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > rhmpropd		GIF version

Theorem rhmpropd 13753

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 2193	. . . . . 6 ⊢ (mulGrp‘𝐽) = (mulGrp‘𝐽)
2		eqid 2193	. . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
3	1, 2	isrhm 13657	. . . . 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 2193	. . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
7		eqid 2193	. . . . . 6 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
8	6, 7	isrhm 13657	. . . . 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 13537	. . . . 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 13537	. . . . 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 13356	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
24	23	eleq2d 2263	. . . . . . . 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 2193	. . . . . . . . . . . 12 ⊢ (Base‘𝐽) = (Base‘𝐽)
29	1, 28	mgpbasg 13425	. . . . . . . . . . 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 2226	. . . . . . . . 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 2193	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
35	2, 34	mgpbasg 13425	. . . . . . . . . . 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 2226	. . . . . . . . 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 2193	. . . . . . . . . . . 12 ⊢ (Base‘𝐿) = (Base‘𝐿)
41	6, 40	mgpbasg 13425	. . . . . . . . . . 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 2226	. . . . . . . . 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 2193	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
47	7, 46	mgpbasg 13425	. . . . . . . . . . 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 2226	. . . . . . . . 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 2193	. . . . . . . . . . . . 13 ⊢ (._r‘𝐽) = (._r‘𝐽)
53	1, 52	mgpplusgg 13423	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Ring → (._r‘𝐽) = (+_g‘(mulGrp‘𝐽)))
54	53	oveqd 5936	. . . . . . . . . . 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 2193	. . . . . . . . . . . . 13 ⊢ (._r‘𝐿) = (._r‘𝐿)
58	6, 57	mgpplusgg 13423	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Ring → (._r‘𝐿) = (+_g‘(mulGrp‘𝐿)))
59	58	oveqd 5936	. . . . . . . . . . 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 2234	. . . . . . . . 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 2193	. . . . . . . . . . . . 13 ⊢ (._r‘𝐾) = (._r‘𝐾)
65	2, 64	mgpplusgg 13423	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (._r‘𝐾) = (+_g‘(mulGrp‘𝐾)))
66	65	oveqd 5936	. . . . . . . . . . 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 2193	. . . . . . . . . . . . 13 ⊢ (._r‘𝑀) = (._r‘𝑀)
70	7, 69	mgpplusgg 13423	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ Ring → (._r‘𝑀) = (+_g‘(mulGrp‘𝑀)))
71	70	oveqd 5936	. . . . . . . . . . 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 2234	. . . . . . . . 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 13041	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐽 ∈ Ring ∧ 𝐾 ∈ Ring)) → ((mulGrp‘𝐽) MndHom (mulGrp‘𝐾)) = ((mulGrp‘𝐿) MndHom (mulGrp‘𝑀)))
75	74	eleq2d 2263	. . . . . . 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 706	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 RingHom 𝐾) ↔ 𝑓 ∈ (𝐿 RingHom 𝑀)))
81	80	eqrdv 2191	1 ⊢ (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 ‘cfv 5255 (class class class)co 5919 Basecbs 12621 +_gcplusg 12698 ._rcmulr 12699 MndHom cmhm 13032 GrpHom cghm 13313 mulGrpcmgp 13419 Ringcrg 13495 RingHom crh 13649

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-pre-ltirr 7986 ax-pre-ltadd 7990

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-map 6706 df-pnf 8058 df-mnf 8059 df-ltxr 8061 df-inn 8985 df-2 9043 df-3 9044 df-ndx 12624 df-slot 12625 df-base 12627 df-sets 12628 df-plusg 12711 df-mulr 12712 df-0g 12872 df-mgm 12942 df-sgrp 12988 df-mnd 13001 df-mhm 13034 df-grp 13078 df-ghm 13314 df-mgp 13420 df-ur 13459 df-ring 13497 df-rhm 13651

This theorem is referenced by: zrhpropd 14125

W3C validator