Theorem uzlidlring 48517

Description: Only the zero (left) ideal or the unit (left) ideal of a domain is a unital ring. (Contributed by AV, 18-Feb-2020.)

Hypotheses

Ref	Expression
lidlabl.l	⊢ 𝐿 = (LIdeal‘𝑅)
lidlabl.i	⊢ 𝐼 = (𝑅 ↾s 𝑈)
zlidlring.b	⊢ 𝐵 = (Base‘𝑅)
zlidlring.0	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
uzlidlring	⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))

Proof of Theorem uzlidlring

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

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
2		eqid 2737	. . 3 ⊢ (.r‘𝐼) = (.r‘𝐼)
3	1, 2	isringrng 20226	. 2 ⊢ (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
4		domnring 20644	. . . . 5 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)
5	4	anim1i 616	. . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿))
6		lidlabl.l	. . . . 5 ⊢ 𝐿 = (LIdeal‘𝑅)
7		lidlabl.i	. . . . 5 ⊢ 𝐼 = (𝑅 ↾s 𝑈)
8	6, 7	lidlrng 48515	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
9	5, 8	syl 17	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
10		ibar 528	. . . . . 6 ⊢ (𝐼 ∈ Rng → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦))))
11	10	bicomd 223	. . . . 5 ⊢ (𝐼 ∈ Rng → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) ↔ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
12	11	adantl 481	. . . 4 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) ↔ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
13		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (.r‘𝑅) = (.r‘𝑅)
14	7, 13	ressmulr 17231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼))
15	14	eqcomd 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝐼) = (.r‘𝑅))
16	15	oveqd 7377	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → (𝑥(.r‘𝐼)𝑦) = (𝑥(.r‘𝑅)𝑦))
17	16	eqeq1d 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → ((𝑥(.r‘𝐼)𝑦) = 𝑦 ↔ (𝑥(.r‘𝑅)𝑦) = 𝑦))
18	15	oveqd 7377	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → (𝑦(.r‘𝐼)𝑥) = (𝑦(.r‘𝑅)𝑥))
19	18	eqeq1d 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → ((𝑦(.r‘𝐼)𝑥) = 𝑦 ↔ (𝑦(.r‘𝑅)𝑥) = 𝑦))
20	17, 19	anbi12d 633	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ 𝐿 → (((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
21	20	ad2antlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
22	21	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
23	22	ralbidv 3160	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
24		simp-4l 783	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑅 ∈ Domn)
25	6, 7	lidlbas 21173	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
26	25	eleq1d 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) ∈ 𝐿 ↔ 𝑈 ∈ 𝐿))
27	26	ibir 268	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿)
28	27	ad3antlr 732	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ∈ 𝐿)
29	25	ad2antlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (Base‘𝐼) = 𝑈)
30	29	eqeq1d 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } ↔ 𝑈 = { 0 }))
31	30	biimpd 229	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } → 𝑈 = { 0 }))
32	31	necon3bd 2947	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (¬ 𝑈 = { 0 } → (Base‘𝐼) ≠ { 0 }))
33	32	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ≠ { 0 })
34	28, 33	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
35	34	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
36		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ (Base‘𝐼))
37		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑅) = (1r‘𝑅)
38		zlidlring.0	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝑅)
39	6, 13, 37, 38	lidldomn1 48513	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Domn ∧ ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦) → 𝑥 = (1r‘𝑅)))
40	24, 35, 36, 39	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦) → 𝑥 = (1r‘𝑅)))
41	23, 40	sylbid 240	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) → 𝑥 = (1r‘𝑅)))
42	41	imp 406	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) → 𝑥 = (1r‘𝑅))
43	25	ad3antlr 732	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) = 𝑈)
44	43	eleq2d 2823	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) ↔ 𝑥 ∈ 𝑈))
45	44	biimpd 229	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) → 𝑥 ∈ 𝑈))
46	45	imp 406	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ 𝑈)
47	46	adantr 480	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) → 𝑥 ∈ 𝑈)
48	42, 47	eqeltrrd 2838	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) → (1r‘𝑅) ∈ 𝑈)
49	48	rexlimdva2 3140	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) → (1r‘𝑅) ∈ 𝑈))
50	49	impancom 451	. . . . . . . 8 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) → (¬ 𝑈 = { 0 } → (1r‘𝑅) ∈ 𝑈))
51	5	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿))
52		zlidlring.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
53	6, 52, 37	lidl1el 21185	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → ((1r‘𝑅) ∈ 𝑈 ↔ 𝑈 = 𝐵))
54	51, 53	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((1r‘𝑅) ∈ 𝑈 ↔ 𝑈 = 𝐵))
55	54	adantr 480	. . . . . . . 8 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) → ((1r‘𝑅) ∈ 𝑈 ↔ 𝑈 = 𝐵))
56	50, 55	sylibd 239	. . . . . . 7 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) → (¬ 𝑈 = { 0 } → 𝑈 = 𝐵))
57	56	orrd 864	. . . . . 6 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) → (𝑈 = { 0 } ∨ 𝑈 = 𝐵))
58	57	ex 412	. . . . 5 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) → (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
59	6, 7, 52, 38	zlidlring 48516	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
60	3	simprbi 496	. . . . . . . . . 10 ⊢ (𝐼 ∈ Ring → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦))
61	59, 60	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦))
62	61	ex 412	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
63	4, 62	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ Domn → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
64	63	ad2antrr 727	. . . . . 6 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
65	5	anim1i 616	. . . . . . 7 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng))
66	52, 13	ringideu 20193	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
67		reurex 3355	. . . . . . . . . . . 12 ⊢ (∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
68	66, 67	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
69	68	adantr 480	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
70	69	ad2antrr 727	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
71	7, 52	ressbas 17167	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐿 → (𝑈 ∩ 𝐵) = (Base‘𝐼))
72	71	ad3antlr 732	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈 ∩ 𝐵) = (Base‘𝐼))
73		ineq1 4166	. . . . . . . . . . . . 13 ⊢ (𝑈 = 𝐵 → (𝑈 ∩ 𝐵) = (𝐵 ∩ 𝐵))
74		inidm 4180	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐵) = 𝐵
75	73, 74	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑈 = 𝐵 → (𝑈 ∩ 𝐵) = 𝐵)
76	75	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈 ∩ 𝐵) = 𝐵)
77	72, 76	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (Base‘𝐼) = 𝐵)
78	20	ad3antlr 732	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
79	77, 78	raleqbidv 3317	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
80	77, 79	rexeqbidv 3318	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
81	70, 80	mpbird 257	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦))
82	81	ex 412	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = 𝐵 → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
83	65, 82	syl 17	. . . . . 6 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = 𝐵 → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
84	64, 83	jaod 860	. . . . 5 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((𝑈 = { 0 } ∨ 𝑈 = 𝐵) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
85	58, 84	impbid 212	. . . 4 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦) ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
86	12, 85	bitrd 279	. . 3 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
87	9, 86	mpdan 688	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)) ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
88	3, 87	bitrid 283	1 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  ∃!wreu 3349   ∩ cin 3901  {csn 4581  ‘cfv 6493  (class class class)co 7360  Basecbs 17140   ↾_s cress 17161  ._rcmulr 17182  0_gc0g 17363  Rngcrng 20091  1_rcur 20120  Ringcrg 20172  Domncdomn 20629  LIdealclidl 21165

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-sca 17197  df-vsca 17198  df-ip 17199  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18870  df-minusg 18871  df-sbg 18872  df-subg 19057  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-nzr 20450  df-subrg 20507  df-domn 20632  df-lmod 20817  df-lss 20887  df-sra 21129  df-rgmod 21130  df-lidl 21167

This theorem is referenced by:  lidldomnnring  48518