Theorem rhmopp 14406

Description: A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)

Assertion

Ref	Expression
rhmopp	⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))

Proof of Theorem rhmopp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. 2 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
2		eqid 2234	. 2 ⊢ (1r‘(oppr‘𝑅)) = (1r‘(oppr‘𝑅))
3		eqid 2234	. 2 ⊢ (1r‘(oppr‘𝑆)) = (1r‘(oppr‘𝑆))
4		eqid 2234	. 2 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
5		eqid 2234	. 2 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
6		rhmrcl1 14385	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7		eqid 2234	. . . . 5 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
8	7	opprringbg 14308	. . . 4 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Ring ↔ (oppr‘𝑅) ∈ Ring))
9	6, 8	syl 14	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑅 ∈ Ring ↔ (oppr‘𝑅) ∈ Ring))
10	6, 9	mpbid 147	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Ring)
11		rhmrcl2 14386	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
12		eqid 2234	. . . . 5 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
13	12	opprringbg 14308	. . . 4 ⊢ (𝑆 ∈ Ring → (𝑆 ∈ Ring ↔ (oppr‘𝑆) ∈ Ring))
14	11, 13	syl 14	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑆 ∈ Ring ↔ (oppr‘𝑆) ∈ Ring))
15	11, 14	mpbid 147	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Ring)
16		eqid 2234	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
17		eqid 2234	. . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆)
18	16, 17	rhm1 14397	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
19	7, 16	oppr1g 14311	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
20	6, 19	syl 14	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
21	20	eqcomd 2240	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r‘(oppr‘𝑅)) = (1r‘𝑅))
22	21	fveq2d 5679	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr‘𝑅))) = (𝐹‘(1r‘𝑅)))
23	12, 17	oppr1g 14311	. . . . 5 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) = (1r‘(oppr‘𝑆)))
24	11, 23	syl 14	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r‘𝑆) = (1r‘(oppr‘𝑆)))
25	24	eqcomd 2240	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (1r‘(oppr‘𝑆)) = (1r‘𝑆))
26	18, 22, 25	3eqtr4d 2277	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr‘𝑅))) = (1r‘(oppr‘𝑆)))
27		simpl 109	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝐹 ∈ (𝑅 RingHom 𝑆))
28		simprr 533	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘(oppr‘𝑅)))
29		eqid 2234	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	7, 29	opprbasg 14303	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
31	6, 30	syl 14	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
32	27, 31	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
33	28, 32	eleqtrrd 2314	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘𝑅))
34		simprl 531	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘(oppr‘𝑅)))
35	34, 32	eleqtrrd 2314	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘𝑅))
36		eqid 2234	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
37		eqid 2234	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
38	29, 36, 37	rhmmul 14394	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(𝑦(.r‘𝑅)𝑥)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥)))
39	27, 33, 35, 38	syl3anc 1274	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (𝐹‘(𝑦(.r‘𝑅)𝑥)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥)))
40	27, 6	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑅 ∈ Ring)
41	29, 36, 7, 4	opprmulg 14299	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅))) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
42	40, 34, 28, 41	syl3anc 1274	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
43	42	fveq2d 5679	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (𝐹‘(𝑥(.r‘(oppr‘𝑅))𝑦)) = (𝐹‘(𝑦(.r‘𝑅)𝑥)))
44	27, 11	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑆 ∈ Ring)
45		eqid 2234	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
46	29, 45	rhmf 14393	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
47	27, 46	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
48	47, 35	ffvelcdmd 5818	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (𝐹‘𝑥) ∈ (Base‘𝑆))
49	47, 33	ffvelcdmd 5818	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (𝐹‘𝑦) ∈ (Base‘𝑆))
50	45, 37, 12, 5	opprmulg 14299	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘𝑥) ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(.r‘(oppr‘𝑆))(𝐹‘𝑦)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥)))
51	44, 48, 49, 50	syl3anc 1274	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → ((𝐹‘𝑥)(.r‘(oppr‘𝑆))(𝐹‘𝑦)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥)))
52	39, 43, 51	3eqtr4d 2277	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (𝐹‘(𝑥(.r‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(.r‘(oppr‘𝑆))(𝐹‘𝑦)))
53	10	ringgrpd 14233	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Grp)
54	15	ringgrpd 14233	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Grp)
55		rhmghm 14392	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
56	55	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
57		simplr 529	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
58		simpr 110	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
59		eqid 2234	. . . . . . . . . 10 ⊢ (+g‘𝑅) = (+g‘𝑅)
60		eqid 2234	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
61	29, 59, 60	ghmlin 14049	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
62	56, 57, 58, 61	syl3anc 1274	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
63	62	ralrimiva 2617	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
64	63	ralrimiva 2617	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
65	46, 64	jca 306	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))
66	53, 54, 65	jca31 309	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (((oppr‘𝑅) ∈ Grp ∧ (oppr‘𝑆) ∈ Grp) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))
67	12, 45	opprbasg 14303	. . . . . . . 8 ⊢ (𝑆 ∈ Ring → (Base‘𝑆) = (Base‘(oppr‘𝑆)))
68	11, 67	syl 14	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (Base‘𝑆) = (Base‘(oppr‘𝑆)))
69	31, 68	feq23d 5509	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ↔ 𝐹:(Base‘(oppr‘𝑅))⟶(Base‘(oppr‘𝑆))))
70	7, 59	oppraddg 14304	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘(oppr‘𝑅)))
71	6, 70	syl 14	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (+g‘𝑅) = (+g‘(oppr‘𝑅)))
72	71	oveqd 6075	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑅))𝑦))
73	72	fveq2d 5679	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘(𝑥(+g‘(oppr‘𝑅))𝑦)))
74	12, 60	oppraddg 14304	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Ring → (+g‘𝑆) = (+g‘(oppr‘𝑆)))
75	11, 74	syl 14	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (+g‘𝑆) = (+g‘(oppr‘𝑆)))
76	75	oveqd 6075	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘(oppr‘𝑆))(𝐹‘𝑦)))
77	73, 76	eqeq12d 2249	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ↔ (𝐹‘(𝑥(+g‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(+g‘(oppr‘𝑆))(𝐹‘𝑦))))
78	31, 77	raleqbidv 2759	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ↔ ∀𝑦 ∈ (Base‘(oppr‘𝑅))(𝐹‘(𝑥(+g‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(+g‘(oppr‘𝑆))(𝐹‘𝑦))))
79	31, 78	raleqbidv 2759	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ↔ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ (Base‘(oppr‘𝑅))(𝐹‘(𝑥(+g‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(+g‘(oppr‘𝑆))(𝐹‘𝑦))))
80	69, 79	anbi12d 473	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ (𝐹:(Base‘(oppr‘𝑅))⟶(Base‘(oppr‘𝑆)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ (Base‘(oppr‘𝑅))(𝐹‘(𝑥(+g‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(+g‘(oppr‘𝑆))(𝐹‘𝑦)))))
81	80	anbi2d 464	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((((oppr‘𝑅) ∈ Grp ∧ (oppr‘𝑆) ∈ Grp) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))) ↔ (((oppr‘𝑅) ∈ Grp ∧ (oppr‘𝑆) ∈ Grp) ∧ (𝐹:(Base‘(oppr‘𝑅))⟶(Base‘(oppr‘𝑆)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ (Base‘(oppr‘𝑅))(𝐹‘(𝑥(+g‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(+g‘(oppr‘𝑆))(𝐹‘𝑦))))))
82	66, 81	mpbid 147	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (((oppr‘𝑅) ∈ Grp ∧ (oppr‘𝑆) ∈ Grp) ∧ (𝐹:(Base‘(oppr‘𝑅))⟶(Base‘(oppr‘𝑆)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ (Base‘(oppr‘𝑅))(𝐹‘(𝑥(+g‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(+g‘(oppr‘𝑆))(𝐹‘𝑦)))))
83		eqid 2234	. . . 4 ⊢ (Base‘(oppr‘𝑆)) = (Base‘(oppr‘𝑆))
84		eqid 2234	. . . 4 ⊢ (+g‘(oppr‘𝑅)) = (+g‘(oppr‘𝑅))
85		eqid 2234	. . . 4 ⊢ (+g‘(oppr‘𝑆)) = (+g‘(oppr‘𝑆))
86	1, 83, 84, 85	isghm 14044	. . 3 ⊢ (𝐹 ∈ ((oppr‘𝑅) GrpHom (oppr‘𝑆)) ↔ (((oppr‘𝑅) ∈ Grp ∧ (oppr‘𝑆) ∈ Grp) ∧ (𝐹:(Base‘(oppr‘𝑅))⟶(Base‘(oppr‘𝑆)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ (Base‘(oppr‘𝑅))(𝐹‘(𝑥(+g‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(+g‘(oppr‘𝑆))(𝐹‘𝑦)))))
87	82, 86	sylibr 134	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) GrpHom (oppr‘𝑆)))
88	1, 2, 3, 4, 5, 10, 15, 26, 52, 87	isrhm2d 14395	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  ._rcmulr 13375  Grpcgrp 13797   GrpHom cghm 14041  1_rcur 14187  Ringcrg 14224  opp_rcoppr 14295   RingHom crh 14380

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-tpos 6489  df-map 6897  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-mhm 13756  df-grp 13800  df-ghm 14042  df-mgp 14149  df-ur 14188  df-ring 14226  df-oppr 14296  df-rhm 14382

This theorem is referenced by:  elrhmunit  14407