Theorem issrngd 19625

Description: Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)

Hypotheses

Ref	Expression
issrngd.k	⊢ (𝜑 → 𝐾 = (Base‘𝑅))
issrngd.p	⊢ (𝜑 → + = (+g‘𝑅))
issrngd.t	⊢ (𝜑 → · = (.r‘𝑅))
issrngd.c	⊢ (𝜑 → ∗ = (*𝑟‘𝑅))
issrngd.r	⊢ (𝜑 → 𝑅 ∈ Ring)
issrngd.cl	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ( ∗ ‘𝑥) ∈ 𝐾)
issrngd.dp	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 + 𝑦)) = (( ∗ ‘𝑥) + ( ∗ ‘𝑦)))
issrngd.dt	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 · 𝑦)) = (( ∗ ‘𝑦) · ( ∗ ‘𝑥)))
issrngd.id	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ( ∗ ‘( ∗ ‘𝑥)) = 𝑥)

Assertion

Ref	Expression
issrngd	⊢ (𝜑 → 𝑅 ∈ *-Ring)

Distinct variable groups:   𝑥,𝑦,𝐾   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   + (𝑥,𝑦)   · (𝑥,𝑦)   ∗ (𝑥,𝑦)

Proof of Theorem issrngd

Step	Hyp	Ref	Expression
1		eqid 2798	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2798	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
3		eqid 2798	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
4	3, 2	oppr1 19380	. . 3 ⊢ (1r‘𝑅) = (1r‘(oppr‘𝑅))
5		eqid 2798	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2798	. . 3 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
7		issrngd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
8	3	opprring 19377	. . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → (oppr‘𝑅) ∈ Ring)
10		id 22	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → 𝑥 = (1r‘𝑅))
11		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝑅) → ((*𝑟‘𝑅)‘𝑥) = ((*𝑟‘𝑅)‘(1r‘𝑅)))
12	11	fveq2d 6649	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
13	10, 12	eqeq12d 2814	. . . . . . . 8 ⊢ (𝑥 = (1r‘𝑅) → (𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) ↔ (1r‘𝑅) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅)))))
14		issrngd.id	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ( ∗ ‘( ∗ ‘𝑥)) = 𝑥)
15	14	ex 416	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐾 → ( ∗ ‘( ∗ ‘𝑥)) = 𝑥))
16		issrngd.k	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 = (Base‘𝑅))
17	16	eleq2d 2875	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ 𝑥 ∈ (Base‘𝑅)))
18		issrngd.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∗ = (*𝑟‘𝑅))
19	18	fveq1d 6647	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ∗ ‘𝑥) = ((*𝑟‘𝑅)‘𝑥))
20	18, 19	fveq12d 6652	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ∗ ‘( ∗ ‘𝑥)) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
21	20	eqeq1d 2800	. . . . . . . . . . . 12 ⊢ (𝜑 → (( ∗ ‘( ∗ ‘𝑥)) = 𝑥 ↔ ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = 𝑥))
22	15, 17, 21	3imtr3d 296	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = 𝑥))
23	22	imp 410	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = 𝑥)
24	23	eqcomd 2804	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
25	24	ralrimiva 3149	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
26	1, 2	ringidcl 19314	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
27	7, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
28	13, 25, 27	rspcdva 3573	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
29	28	oveq1d 7150	. . . . . 6 ⊢ (𝜑 → ((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))) = (((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅)))(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))))
30	11	eleq1d 2874	. . . . . . . 8 ⊢ (𝑥 = (1r‘𝑅) → (((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅) ↔ ((*𝑟‘𝑅)‘(1r‘𝑅)) ∈ (Base‘𝑅)))
31		issrngd.cl	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ( ∗ ‘𝑥) ∈ 𝐾)
32	31	ex 416	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐾 → ( ∗ ‘𝑥) ∈ 𝐾))
33	19, 16	eleq12d 2884	. . . . . . . . . 10 ⊢ (𝜑 → (( ∗ ‘𝑥) ∈ 𝐾 ↔ ((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅)))
34	32, 17, 33	3imtr3d 296	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅)))
35	34	ralrimiv 3148	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅))
36	30, 35, 27	rspcdva 3573	. . . . . . 7 ⊢ (𝜑 → ((*𝑟‘𝑅)‘(1r‘𝑅)) ∈ (Base‘𝑅))
37		issrngd.dt	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 · 𝑦)) = (( ∗ ‘𝑦) · ( ∗ ‘𝑥)))
38	37	3expib 1119	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 · 𝑦)) = (( ∗ ‘𝑦) · ( ∗ ‘𝑥))))
39	16	eleq2d 2875	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↔ 𝑦 ∈ (Base‘𝑅)))
40	17, 39	anbi12d 633	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
41		issrngd.t	. . . . . . . . . . . 12 ⊢ (𝜑 → · = (.r‘𝑅))
42	41	oveqd 7152	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 · 𝑦) = (𝑥(.r‘𝑅)𝑦))
43	18, 42	fveq12d 6652	. . . . . . . . . 10 ⊢ (𝜑 → ( ∗ ‘(𝑥 · 𝑦)) = ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)))
44	18	fveq1d 6647	. . . . . . . . . . 11 ⊢ (𝜑 → ( ∗ ‘𝑦) = ((*𝑟‘𝑅)‘𝑦))
45	41, 44, 19	oveq123d 7156	. . . . . . . . . 10 ⊢ (𝜑 → (( ∗ ‘𝑦) · ( ∗ ‘𝑥)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥)))
46	43, 45	eqeq12d 2814	. . . . . . . . 9 ⊢ (𝜑 → (( ∗ ‘(𝑥 · 𝑦)) = (( ∗ ‘𝑦) · ( ∗ ‘𝑥)) ↔ ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥))))
47	38, 40, 46	3imtr3d 296	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥))))
48	47	ralrimivv 3155	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥)))
49		fvoveq1 7158	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)𝑦)))
50	11	oveq2d 7151	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))))
51	49, 50	eqeq12d 2814	. . . . . . . 8 ⊢ (𝑥 = (1r‘𝑅) → (((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥)) ↔ ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))))
52		oveq2 7143	. . . . . . . . . 10 ⊢ (𝑦 = ((*𝑟‘𝑅)‘(1r‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑦) = ((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))))
53	52	fveq2d 6649	. . . . . . . . 9 ⊢ (𝑦 = ((*𝑟‘𝑅)‘(1r‘𝑅)) → ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)𝑦)) = ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))))
54		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑦 = ((*𝑟‘𝑅)‘(1r‘𝑅)) → ((*𝑟‘𝑅)‘𝑦) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
55	54	oveq1d 7150	. . . . . . . . 9 ⊢ (𝑦 = ((*𝑟‘𝑅)‘(1r‘𝑅)) → (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))) = (((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅)))(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))))
56	53, 55	eqeq12d 2814	. . . . . . . 8 ⊢ (𝑦 = ((*𝑟‘𝑅)‘(1r‘𝑅)) → (((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))) ↔ ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))) = (((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅)))(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))))
57	51, 56	rspc2va 3582	. . . . . . 7 ⊢ ((((1r‘𝑅) ∈ (Base‘𝑅) ∧ ((*𝑟‘𝑅)‘(1r‘𝑅)) ∈ (Base‘𝑅)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥))) → ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))) = (((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅)))(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))))
58	27, 36, 48, 57	syl21anc 836	. . . . . 6 ⊢ (𝜑 → ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))) = (((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅)))(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))))
59	29, 58	eqtr4d 2836	. . . . 5 ⊢ (𝜑 → ((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))) = ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))))
60	1, 5, 2	ringlidm 19317	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((*𝑟‘𝑅)‘(1r‘𝑅)) ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))) = ((*𝑟‘𝑅)‘(1r‘𝑅)))
61	7, 36, 60	syl2anc 587	. . . . 5 ⊢ (𝜑 → ((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))) = ((*𝑟‘𝑅)‘(1r‘𝑅)))
62	61	fveq2d 6649	. . . . 5 ⊢ (𝜑 → ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
63	59, 61, 62	3eqtr3d 2841	. . . 4 ⊢ (𝜑 → ((*𝑟‘𝑅)‘(1r‘𝑅)) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
64		eqid 2798	. . . . . 6 ⊢ (*𝑟‘𝑅) = (*𝑟‘𝑅)
65		eqid 2798	. . . . . 6 ⊢ (*rf‘𝑅) = (*rf‘𝑅)
66	1, 64, 65	stafval 19612	. . . . 5 ⊢ ((1r‘𝑅) ∈ (Base‘𝑅) → ((*rf‘𝑅)‘(1r‘𝑅)) = ((*𝑟‘𝑅)‘(1r‘𝑅)))
67	27, 66	syl 17	. . . 4 ⊢ (𝜑 → ((*rf‘𝑅)‘(1r‘𝑅)) = ((*𝑟‘𝑅)‘(1r‘𝑅)))
68	63, 67, 28	3eqtr4d 2843	. . 3 ⊢ (𝜑 → ((*rf‘𝑅)‘(1r‘𝑅)) = (1r‘𝑅))
69	47	imp 410	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥)))
70	1, 5, 3, 6	opprmul 19372	. . . . 5 ⊢ (((*𝑟‘𝑅)‘𝑥)(.r‘(oppr‘𝑅))((*𝑟‘𝑅)‘𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥))
71	69, 70	eqtr4di 2851	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑥)(.r‘(oppr‘𝑅))((*𝑟‘𝑅)‘𝑦)))
72	1, 5	ringcl 19307	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
73	72	3expb 1117	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
74	7, 73	sylan 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
75	1, 64, 65	stafval 19612	. . . . 5 ⊢ ((𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅) → ((*rf‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)))
76	74, 75	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)))
77	1, 64, 65	stafval 19612	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑅) → ((*rf‘𝑅)‘𝑥) = ((*𝑟‘𝑅)‘𝑥))
78	1, 64, 65	stafval 19612	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑅) → ((*rf‘𝑅)‘𝑦) = ((*𝑟‘𝑅)‘𝑦))
79	77, 78	oveqan12d 7154	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (((*rf‘𝑅)‘𝑥)(.r‘(oppr‘𝑅))((*rf‘𝑅)‘𝑦)) = (((*𝑟‘𝑅)‘𝑥)(.r‘(oppr‘𝑅))((*𝑟‘𝑅)‘𝑦)))
80	79	adantl 485	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (((*rf‘𝑅)‘𝑥)(.r‘(oppr‘𝑅))((*rf‘𝑅)‘𝑦)) = (((*𝑟‘𝑅)‘𝑥)(.r‘(oppr‘𝑅))((*𝑟‘𝑅)‘𝑦)))
81	71, 76, 80	3eqtr4d 2843	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*rf‘𝑅)‘𝑥)(.r‘(oppr‘𝑅))((*rf‘𝑅)‘𝑦)))
82	3, 1	opprbas 19375	. . 3 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
83		eqid 2798	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
84	3, 83	oppradd 19376	. . 3 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅))
85	34	imp 410	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅))
86	1, 64, 65	staffval 19611	. . . 4 ⊢ (*rf‘𝑅) = (𝑥 ∈ (Base‘𝑅) ↦ ((*𝑟‘𝑅)‘𝑥))
87	85, 86	fmptd 6855	. . 3 ⊢ (𝜑 → (*rf‘𝑅):(Base‘𝑅)⟶(Base‘𝑅))
88		issrngd.dp	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 + 𝑦)) = (( ∗ ‘𝑥) + ( ∗ ‘𝑦)))
89	88	3expib 1119	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 + 𝑦)) = (( ∗ ‘𝑥) + ( ∗ ‘𝑦))))
90		issrngd.p	. . . . . . . . 9 ⊢ (𝜑 → + = (+g‘𝑅))
91	90	oveqd 7152	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
92	18, 91	fveq12d 6652	. . . . . . 7 ⊢ (𝜑 → ( ∗ ‘(𝑥 + 𝑦)) = ((*𝑟‘𝑅)‘(𝑥(+g‘𝑅)𝑦)))
93	90, 19, 44	oveq123d 7156	. . . . . . 7 ⊢ (𝜑 → (( ∗ ‘𝑥) + ( ∗ ‘𝑦)) = (((*𝑟‘𝑅)‘𝑥)(+g‘𝑅)((*𝑟‘𝑅)‘𝑦)))
94	92, 93	eqeq12d 2814	. . . . . 6 ⊢ (𝜑 → (( ∗ ‘(𝑥 + 𝑦)) = (( ∗ ‘𝑥) + ( ∗ ‘𝑦)) ↔ ((*𝑟‘𝑅)‘(𝑥(+g‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑥)(+g‘𝑅)((*𝑟‘𝑅)‘𝑦))))
95	89, 40, 94	3imtr3d 296	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟‘𝑅)‘(𝑥(+g‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑥)(+g‘𝑅)((*𝑟‘𝑅)‘𝑦))))
96	95	imp 410	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*𝑟‘𝑅)‘(𝑥(+g‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑥)(+g‘𝑅)((*𝑟‘𝑅)‘𝑦)))
97	1, 83	ringacl 19324	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
98	97	3expb 1117	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
99	7, 98	sylan 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
100	1, 64, 65	stafval 19612	. . . . 5 ⊢ ((𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅) → ((*rf‘𝑅)‘(𝑥(+g‘𝑅)𝑦)) = ((*𝑟‘𝑅)‘(𝑥(+g‘𝑅)𝑦)))
101	99, 100	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf‘𝑅)‘(𝑥(+g‘𝑅)𝑦)) = ((*𝑟‘𝑅)‘(𝑥(+g‘𝑅)𝑦)))
102	77, 78	oveqan12d 7154	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (((*rf‘𝑅)‘𝑥)(+g‘𝑅)((*rf‘𝑅)‘𝑦)) = (((*𝑟‘𝑅)‘𝑥)(+g‘𝑅)((*𝑟‘𝑅)‘𝑦)))
103	102	adantl 485	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (((*rf‘𝑅)‘𝑥)(+g‘𝑅)((*rf‘𝑅)‘𝑦)) = (((*𝑟‘𝑅)‘𝑥)(+g‘𝑅)((*𝑟‘𝑅)‘𝑦)))
104	96, 101, 103	3eqtr4d 2843	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((*rf‘𝑅)‘(𝑥(+g‘𝑅)𝑦)) = (((*rf‘𝑅)‘𝑥)(+g‘𝑅)((*rf‘𝑅)‘𝑦)))
105	1, 2, 4, 5, 6, 7, 9, 68, 81, 82, 83, 84, 87, 104	isrhmd 19477	. 2 ⊢ (𝜑 → (*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)))
106	1, 64, 65	staffval 19611	. . 3 ⊢ (*rf‘𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟‘𝑅)‘𝑦))
107	106	fmpt 6851	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘𝑦) ∈ (Base‘𝑅) ↔ (*rf‘𝑅):(Base‘𝑅)⟶(Base‘𝑅))
108	87, 107	sylibr 237	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘𝑦) ∈ (Base‘𝑅))
109	108	r19.21bi 3173	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟‘𝑅)‘𝑦) ∈ (Base‘𝑅))
110		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
111		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((*𝑟‘𝑅)‘𝑥) = ((*𝑟‘𝑅)‘𝑦))
112	111	fveq2d 6649	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
113	110, 112	eqeq12d 2814	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) ↔ 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦))))
114	113	rspccva 3570	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
115	25, 114	sylan 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
116	115	adantrl 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
117		fveq2 6645	. . . . . . . 8 ⊢ (𝑥 = ((*𝑟‘𝑅)‘𝑦) → ((*𝑟‘𝑅)‘𝑥) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
118	117	eqeq2d 2809	. . . . . . 7 ⊢ (𝑥 = ((*𝑟‘𝑅)‘𝑦) → (𝑦 = ((*𝑟‘𝑅)‘𝑥) ↔ 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦))))
119	116, 118	syl5ibrcom 250	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟‘𝑅)‘𝑦) → 𝑦 = ((*𝑟‘𝑅)‘𝑥)))
120	24	adantrr 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
121		fveq2 6645	. . . . . . . 8 ⊢ (𝑦 = ((*𝑟‘𝑅)‘𝑥) → ((*𝑟‘𝑅)‘𝑦) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
122	121	eqeq2d 2809	. . . . . . 7 ⊢ (𝑦 = ((*𝑟‘𝑅)‘𝑥) → (𝑥 = ((*𝑟‘𝑅)‘𝑦) ↔ 𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥))))
123	120, 122	syl5ibrcom 250	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦 = ((*𝑟‘𝑅)‘𝑥) → 𝑥 = ((*𝑟‘𝑅)‘𝑦)))
124	119, 123	impbid 215	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟‘𝑅)‘𝑦) ↔ 𝑦 = ((*𝑟‘𝑅)‘𝑥)))
125	86, 85, 109, 124	f1ocnv2d 7378	. . . 4 ⊢ (𝜑 → ((*rf‘𝑅):(Base‘𝑅)–1-1-onto→(Base‘𝑅) ∧ ◡(*rf‘𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟‘𝑅)‘𝑦))))
126	125	simprd 499	. . 3 ⊢ (𝜑 → ◡(*rf‘𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟‘𝑅)‘𝑦)))
127	106, 126	eqtr4id 2852	. 2 ⊢ (𝜑 → (*rf‘𝑅) = ◡(*rf‘𝑅))
128	3, 65	issrng 19614	. 2 ⊢ (𝑅 ∈ *-Ring ↔ ((*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)) ∧ (*rf‘𝑅) = ◡(*rf‘𝑅)))
129	105, 127, 128	sylanbrc 586	1 ⊢ (𝜑 → 𝑅 ∈ *-Ring)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ↦ cmpt 5110  ^◡ccnv 5518  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  ._rcmulr 16558  *_𝑟cstv 16559  1_rcur 19244  Ringcrg 19290  opp_rcoppr 19368   RingHom crh 19460  *_rfcstf 19607  *-Ringcsr 19608

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-tpos 7875  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-mulr 16571  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-mgp 19233  df-ur 19245  df-ring 19292  df-oppr 19369  df-rnghom 19463  df-staf 19609  df-srng 19610

This theorem is referenced by:  idsrngd  19626  cnsrng  20125  hlhilsrnglem  39249