Theorem uzlidlring 42777

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 2826	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
2		eqid 2826	. . 3 ⊢ (.r‘𝐼) = (.r‘𝐼)
3	1, 2	isringrng 42729	. 2 ⊢ (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
4		domnring 19658	. . . . 5 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)
5	4	anim1i 610	. . . 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 42775	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
9	5, 8	syl 17	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
10		ibar 526	. . . . . 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 215	. . . . 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 475	. . . 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 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (.r‘𝑅) = (.r‘𝑅)
14	7, 13	ressmulr 16366	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼))
15	14	eqcomd 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝐼) = (.r‘𝑅))
16	15	oveqd 6923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → (𝑥(.r‘𝐼)𝑦) = (𝑥(.r‘𝑅)𝑦))
17	16	eqeq1d 2828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → ((𝑥(.r‘𝐼)𝑦) = 𝑦 ↔ (𝑥(.r‘𝑅)𝑦) = 𝑦))
18	15	oveqd 6923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → (𝑦(.r‘𝐼)𝑥) = (𝑦(.r‘𝑅)𝑥))
19	18	eqeq1d 2828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → ((𝑦(.r‘𝐼)𝑥) = 𝑦 ↔ (𝑦(.r‘𝑅)𝑥) = 𝑦))
20	17, 19	anbi12d 626	. . . . . . . . . . . . . . . 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 720	. . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . 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 3196	. . . . . . . . . . . . 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 803	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑅 ∈ Domn)
25	6, 7	lidlbas 42771	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
26	25	eleq1d 2892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) ∈ 𝐿 ↔ 𝑈 ∈ 𝐿))
27	26	ibir 260	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿)
28	27	ad3antlr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ∈ 𝐿)
29	25	ad2antlr 720	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (Base‘𝐼) = 𝑈)
30	29	eqeq1d 2828	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } ↔ 𝑈 = { 0 }))
31	30	biimpd 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } → 𝑈 = { 0 }))
32	31	necon3bd 3014	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (¬ 𝑈 = { 0 } → (Base‘𝐼) ≠ { 0 }))
33	32	imp 397	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ≠ { 0 })
34	28, 33	jca 509	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
35	34	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
36		simpr 479	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ (Base‘𝐼))
37		eqid 2826	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑅) = (1r‘𝑅)
38		zlidlring.0	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝑅)
39	6, 13, 37, 38	lidldomn1 42769	. . . . . . . . . . . . . 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 1496	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 397	. . . . . . . . . . 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 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) = 𝑈)
44	43	eleq2d 2893	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) ↔ 𝑥 ∈ 𝑈))
45	44	biimpd 221	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) → 𝑥 ∈ 𝑈))
46	45	imp 397	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ 𝑈)
47	46	adantr 474	. . . . . . . . . . 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 2908	. . . . . . . . . 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 3244	. . . . . . . . 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 445	. . . . . . . 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 474	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿))
52		zlidlring.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
53	6, 52, 37	lidl1el 19580	. . . . . . . . . 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 474	. . . . . . . 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 231	. . . . . . 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 896	. . . . . 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 403	. . . . 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 42776	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
60	3	simprbi 492	. . . . . . . . . 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 403	. . . . . . . 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 719	. . . . . 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 610	. . . . . . 7 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng))
66	52, 13	ringideu 18920	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
67		reurex 3373	. . . . . . . . . . . 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 474	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
70	69	ad2antrr 719	. . . . . . . . 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 16294	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐿 → (𝑈 ∩ 𝐵) = (Base‘𝐼))
72	71	ad3antlr 724	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈 ∩ 𝐵) = (Base‘𝐼))
73		ineq1 4035	. . . . . . . . . . . . 13 ⊢ (𝑈 = 𝐵 → (𝑈 ∩ 𝐵) = (𝐵 ∩ 𝐵))
74		inidm 4048	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐵) = 𝐵
75	73, 74	syl6eq 2878	. . . . . . . . . . . 12 ⊢ (𝑈 = 𝐵 → (𝑈 ∩ 𝐵) = 𝐵)
76	75	adantl 475	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈 ∩ 𝐵) = 𝐵)
77	72, 76	eqtr3d 2864	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (Base‘𝐼) = 𝐵)
78	20	ad3antlr 724	. . . . . . . . . . 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 3365	. . . . . . . . . 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 3366	. . . . . . . . 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 249	. . . . . . . 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 403	. . . . . . 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 892	. . . . 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 204	. . . 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 271	. . 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 680	. 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	syl5bb 275	1 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166   ≠ wne 3000  ∀wral 3118  ∃wrex 3119  ∃!wreu 3120   ∩ cin 3798  {csn 4398  ‘cfv 6124  (class class class)co 6906  Basecbs 16223   ↾_s cress 16224  ._rcmulr 16307  0_gc0g 16454  1_rcur 18856  Ringcrg 18902  LIdealclidl 19532  Domncdomn 19642  Rngcrng 42722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-er 8010  df-en 8224  df-dom 8225  df-sdom 8226  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-3 11416  df-4 11417  df-5 11418  df-6 11419  df-7 11420  df-8 11421  df-ndx 16226  df-slot 16227  df-base 16229  df-sets 16230  df-ress 16231  df-plusg 16319  df-mulr 16320  df-sca 16322  df-vsca 16323  df-ip 16324  df-0g 16456  df-mgm 17596  df-sgrp 17638  df-mnd 17649  df-grp 17780  df-minusg 17781  df-sbg 17782  df-subg 17943  df-cmn 18549  df-abl 18550  df-mgp 18845  df-ur 18857  df-ring 18904  df-subrg 19135  df-lmod 19222  df-lss 19290  df-sra 19534  df-rgmod 19535  df-lidl 19536  df-nzr 19620  df-domn 19646  df-rng0 42723

This theorem is referenced by:  lidldomnnring  42778