Theorem issrngd 20612

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 2732	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2732	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
3		eqid 2732	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
4	3, 2	oppr1 20241	. . 3 ⊢ (1r‘𝑅) = (1r‘(oppr‘𝑅))
5		eqid 2732	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2732	. . 3 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
7		issrngd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
8	3	opprring 20238	. . . 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 6891	. . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝑅) → ((*𝑟‘𝑅)‘𝑥) = ((*𝑟‘𝑅)‘(1r‘𝑅)))
12	11	fveq2d 6895	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
13	10, 12	eqeq12d 2748	. . . . . . . 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 413	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐾 → ( ∗ ‘( ∗ ‘𝑥)) = 𝑥))
16		issrngd.k	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 = (Base‘𝑅))
17	16	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ 𝑥 ∈ (Base‘𝑅)))
18		issrngd.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∗ = (*𝑟‘𝑅))
19	18	fveq1d 6893	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ∗ ‘𝑥) = ((*𝑟‘𝑅)‘𝑥))
20	18, 19	fveq12d 6898	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ∗ ‘( ∗ ‘𝑥)) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
21	20	eqeq1d 2734	. . . . . . . . . . . 12 ⊢ (𝜑 → (( ∗ ‘( ∗ ‘𝑥)) = 𝑥 ↔ ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = 𝑥))
22	15, 17, 21	3imtr3d 292	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = 𝑥))
23	22	imp 407	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = 𝑥)
24	23	eqcomd 2738	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
25	24	ralrimiva 3146	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
26	1, 2	ringidcl 20154	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
27	7, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
28	13, 25, 27	rspcdva 3613	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
29	28	oveq1d 7426	. . . . . 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 2818	. . . . . . . 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 413	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐾 → ( ∗ ‘𝑥) ∈ 𝐾))
33	19, 16	eleq12d 2827	. . . . . . . . . 10 ⊢ (𝜑 → (( ∗ ‘𝑥) ∈ 𝐾 ↔ ((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅)))
34	32, 17, 33	3imtr3d 292	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) → ((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅)))
35	34	ralrimiv 3145	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅))
36	30, 35, 27	rspcdva 3613	. . . . . . 7 ⊢ (𝜑 → ((*𝑟‘𝑅)‘(1r‘𝑅)) ∈ (Base‘𝑅))
37		issrngd.dt	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 · 𝑦)) = (( ∗ ‘𝑦) · ( ∗ ‘𝑥)))
38	37	3expib 1122	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 · 𝑦)) = (( ∗ ‘𝑦) · ( ∗ ‘𝑥))))
39	16	eleq2d 2819	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↔ 𝑦 ∈ (Base‘𝑅)))
40	17, 39	anbi12d 631	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
41		issrngd.t	. . . . . . . . . . . 12 ⊢ (𝜑 → · = (.r‘𝑅))
42	41	oveqd 7428	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 · 𝑦) = (𝑥(.r‘𝑅)𝑦))
43	18, 42	fveq12d 6898	. . . . . . . . . 10 ⊢ (𝜑 → ( ∗ ‘(𝑥 · 𝑦)) = ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)))
44	18	fveq1d 6893	. . . . . . . . . . 11 ⊢ (𝜑 → ( ∗ ‘𝑦) = ((*𝑟‘𝑅)‘𝑦))
45	41, 44, 19	oveq123d 7432	. . . . . . . . . 10 ⊢ (𝜑 → (( ∗ ‘𝑦) · ( ∗ ‘𝑥)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥)))
46	43, 45	eqeq12d 2748	. . . . . . . . 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 292	. . . . . . . 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 3198	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥)))
49		fvoveq1 7434	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → ((*𝑟‘𝑅)‘(𝑥(.r‘𝑅)𝑦)) = ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)𝑦)))
50	11	oveq2d 7427	. . . . . . . . 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 2748	. . . . . . . 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 7419	. . . . . . . . . 10 ⊢ (𝑦 = ((*𝑟‘𝑅)‘(1r‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑦) = ((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))))
53	52	fveq2d 6895	. . . . . . . . 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 6891	. . . . . . . . . 10 ⊢ (𝑦 = ((*𝑟‘𝑅)‘(1r‘𝑅)) → ((*𝑟‘𝑅)‘𝑦) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
55	54	oveq1d 7426	. . . . . . . . 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 2748	. . . . . . . 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 3623	. . . . . . 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 2775	. . . . 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 20157	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅))) = ((*𝑟‘𝑅)‘(1r‘𝑅)))
62	61	fveq2d 6895	. . . . 5 ⊢ (𝜑 → ((*𝑟‘𝑅)‘((1r‘𝑅)(.r‘𝑅)((*𝑟‘𝑅)‘(1r‘𝑅)))) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
63	59, 61, 62	3eqtr3d 2780	. . . 4 ⊢ (𝜑 → ((*𝑟‘𝑅)‘(1r‘𝑅)) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘(1r‘𝑅))))
64		eqid 2732	. . . . . 6 ⊢ (*𝑟‘𝑅) = (*𝑟‘𝑅)
65		eqid 2732	. . . . . 6 ⊢ (*rf‘𝑅) = (*rf‘𝑅)
66	1, 64, 65	stafval 20599	. . . . 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 2782	. . 3 ⊢ (𝜑 → ((*rf‘𝑅)‘(1r‘𝑅)) = (1r‘𝑅))
69	47	imp 407	. . . . 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 20228	. . . . 5 ⊢ (((*𝑟‘𝑅)‘𝑥)(.r‘(oppr‘𝑅))((*𝑟‘𝑅)‘𝑦)) = (((*𝑟‘𝑅)‘𝑦)(.r‘𝑅)((*𝑟‘𝑅)‘𝑥))
71	69, 70	eqtr4di 2790	. . . 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 20144	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
73	72	3expb 1120	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
74	7, 73	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
75	1, 64, 65	stafval 20599	. . . . 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 20599	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑅) → ((*rf‘𝑅)‘𝑥) = ((*𝑟‘𝑅)‘𝑥))
78	1, 64, 65	stafval 20599	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑅) → ((*rf‘𝑅)‘𝑦) = ((*𝑟‘𝑅)‘𝑦))
79	77, 78	oveqan12d 7430	. . . . 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 482	. . . 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 2782	. . 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 20232	. . 3 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
83		eqid 2732	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
84	3, 83	oppradd 20234	. . 3 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅))
85	34	imp 407	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((*𝑟‘𝑅)‘𝑥) ∈ (Base‘𝑅))
86	1, 64, 65	staffval 20598	. . . 4 ⊢ (*rf‘𝑅) = (𝑥 ∈ (Base‘𝑅) ↦ ((*𝑟‘𝑅)‘𝑥))
87	85, 86	fmptd 7115	. . 3 ⊢ (𝜑 → (*rf‘𝑅):(Base‘𝑅)⟶(Base‘𝑅))
88		issrngd.dp	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 + 𝑦)) = (( ∗ ‘𝑥) + ( ∗ ‘𝑦)))
89	88	3expib 1122	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 + 𝑦)) = (( ∗ ‘𝑥) + ( ∗ ‘𝑦))))
90		issrngd.p	. . . . . . . . 9 ⊢ (𝜑 → + = (+g‘𝑅))
91	90	oveqd 7428	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
92	18, 91	fveq12d 6898	. . . . . . 7 ⊢ (𝜑 → ( ∗ ‘(𝑥 + 𝑦)) = ((*𝑟‘𝑅)‘(𝑥(+g‘𝑅)𝑦)))
93	90, 19, 44	oveq123d 7432	. . . . . . 7 ⊢ (𝜑 → (( ∗ ‘𝑥) + ( ∗ ‘𝑦)) = (((*𝑟‘𝑅)‘𝑥)(+g‘𝑅)((*𝑟‘𝑅)‘𝑦)))
94	92, 93	eqeq12d 2748	. . . . . 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 292	. . . . 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 407	. . . 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 20166	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
98	97	3expb 1120	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
99	7, 98	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
100	1, 64, 65	stafval 20599	. . . . 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 7430	. . . . 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 482	. . . 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 2782	. . 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 20379	. 2 ⊢ (𝜑 → (*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)))
106	1, 64, 65	staffval 20598	. . 3 ⊢ (*rf‘𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟‘𝑅)‘𝑦))
107	106	fmpt 7111	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘𝑦) ∈ (Base‘𝑅) ↔ (*rf‘𝑅):(Base‘𝑅)⟶(Base‘𝑅))
108	87, 107	sylibr 233	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑅)((*𝑟‘𝑅)‘𝑦) ∈ (Base‘𝑅))
109	108	r19.21bi 3248	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → ((*𝑟‘𝑅)‘𝑦) ∈ (Base‘𝑅))
110		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
111		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((*𝑟‘𝑅)‘𝑥) = ((*𝑟‘𝑅)‘𝑦))
112	111	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
113	110, 112	eqeq12d 2748	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) ↔ 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦))))
114	113	rspccva 3611	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ (Base‘𝑅)𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
115	25, 114	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
116	115	adantrl 714	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
117		fveq2 6891	. . . . . . . 8 ⊢ (𝑥 = ((*𝑟‘𝑅)‘𝑦) → ((*𝑟‘𝑅)‘𝑥) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦)))
118	117	eqeq2d 2743	. . . . . . 7 ⊢ (𝑥 = ((*𝑟‘𝑅)‘𝑦) → (𝑦 = ((*𝑟‘𝑅)‘𝑥) ↔ 𝑦 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑦))))
119	116, 118	syl5ibrcom 246	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟‘𝑅)‘𝑦) → 𝑦 = ((*𝑟‘𝑅)‘𝑥)))
120	24	adantrr 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
121		fveq2 6891	. . . . . . . 8 ⊢ (𝑦 = ((*𝑟‘𝑅)‘𝑥) → ((*𝑟‘𝑅)‘𝑦) = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥)))
122	121	eqeq2d 2743	. . . . . . 7 ⊢ (𝑦 = ((*𝑟‘𝑅)‘𝑥) → (𝑥 = ((*𝑟‘𝑅)‘𝑦) ↔ 𝑥 = ((*𝑟‘𝑅)‘((*𝑟‘𝑅)‘𝑥))))
123	120, 122	syl5ibrcom 246	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦 = ((*𝑟‘𝑅)‘𝑥) → 𝑥 = ((*𝑟‘𝑅)‘𝑦)))
124	119, 123	impbid 211	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 = ((*𝑟‘𝑅)‘𝑦) ↔ 𝑦 = ((*𝑟‘𝑅)‘𝑥)))
125	86, 85, 109, 124	f1ocnv2d 7661	. . . 4 ⊢ (𝜑 → ((*rf‘𝑅):(Base‘𝑅)–1-1-onto→(Base‘𝑅) ∧ ◡(*rf‘𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟‘𝑅)‘𝑦))))
126	125	simprd 496	. . 3 ⊢ (𝜑 → ◡(*rf‘𝑅) = (𝑦 ∈ (Base‘𝑅) ↦ ((*𝑟‘𝑅)‘𝑦)))
127	106, 126	eqtr4id 2791	. 2 ⊢ (𝜑 → (*rf‘𝑅) = ◡(*rf‘𝑅))
128	3, 65	issrng 20601	. 2 ⊢ (𝑅 ∈ *-Ring ↔ ((*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)) ∧ (*rf‘𝑅) = ◡(*rf‘𝑅)))
129	105, 127, 128	sylanbrc 583	1 ⊢ (𝜑 → 𝑅 ∈ *-Ring)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ↦ cmpt 5231  ^◡ccnv 5675  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7411  Basecbs 17148  +_gcplusg 17201  ._rcmulr 17202  *_𝑟cstv 17203  1_rcur 20075  Ringcrg 20127  opp_rcoppr 20224   RingHom crh 20360  *_rfcstf 20594  *-Ringcsr 20595

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

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-op 4635  df-uni 4909  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-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-plusg 17214  df-mulr 17215  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-grp 18858  df-minusg 18859  df-ghm 19128  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-oppr 20225  df-rhm 20363  df-staf 20596  df-srng 20597

This theorem is referenced by:  idsrngd  20613  cnsrng  21179  hlhilsrnglem  41131