Theorem uzlidlring 49425

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 2760	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
2		eqid 2760	. . 3 ⊢ (.r‘𝐼) = (.r‘𝐼)
3	1, 2	isringrng 20521	. 2 ⊢ (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
4		domnring 20964	. . . . 5 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)
5	4	anim1i 627	. . . 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 49423	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
9	5, 8	syl 18	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
10		ibar 538	. . . . . 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 226	. . . . 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 487	. . . 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 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (.r‘𝑅) = (.r‘𝑅)
14	7, 13	ressmulr 17483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼))
15	14	eqcomd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝐼) = (.r‘𝑅))
16	15	oveqd 7437	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → (𝑥(.r‘𝐼)𝑦) = (𝑥(.r‘𝑅)𝑦))
17	16	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → ((𝑥(.r‘𝐼)𝑦) = 𝑦 ↔ (𝑥(.r‘𝑅)𝑦) = 𝑦))
18	15	oveqd 7437	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → (𝑦(.r‘𝐼)𝑥) = (𝑦(.r‘𝑅)𝑥))
19	18	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → ((𝑦(.r‘𝐼)𝑥) = 𝑦 ↔ (𝑦(.r‘𝑅)𝑥) = 𝑦))
20	17, 19	anbi12d 644	. . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . 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 795	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑅 ∈ Domn)
25	6, 7	lidlbas 21498	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
26	25	eleq1d 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) ∈ 𝐿 ↔ 𝑈 ∈ 𝐿))
27	26	ibir 271	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿)
28	27	ad3antlr 744	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ∈ 𝐿)
29	25	ad2antlr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (Base‘𝐼) = 𝑈)
30	29	eqeq1d 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } ↔ 𝑈 = { 0 }))
31	30	biimpd 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } → 𝑈 = { 0 }))
32	31	necon3bd 2969	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (¬ 𝑈 = { 0 } → (Base‘𝐼) ≠ { 0 }))
33	32	imp 412	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ≠ { 0 })
34	28, 33	jca 521	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
35	34	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
36		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ (Base‘𝐼))
37		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑅) = (1r‘𝑅)
38		zlidlring.0	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝑅)
39	6, 13, 37, 38	lidldomn1 49421	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 744	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) = 𝑈)
44	43	eleq2d 2846	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) ↔ 𝑥 ∈ 𝑈))
45	44	biimpd 232	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) → 𝑥 ∈ 𝑈))
46	45	imp 412	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ 𝑈)
47	46	adantr 486	. . . . . . . . . . 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 2861	. . . . . . . . . 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 3165	. . . . . . . . 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 457	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿))
52		zlidlring.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
53	6, 52, 37	lidl1el 21510	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → ((1r‘𝑅) ∈ 𝑈 ↔ 𝑈 = 𝐵))
54	51, 53	syl 18	. . . . . . . . 9 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((1r‘𝑅) ∈ 𝑈 ↔ 𝑈 = 𝐵))
55	54	adantr 486	. . . . . . . 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 242	. . . . . . 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 877	. . . . . 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 418	. . . . 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 49424	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
60	3	simprbi 503	. . . . . . . . . 10 ⊢ (𝐼 ∈ Ring → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦))
61	59, 60	syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦))
62	61	ex 418	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝑅 ∈ Domn → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
64	63	ad2antrr 739	. . . . . 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 627	. . . . . . 7 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng))
66	52, 13	ringideu 20486	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
67		reurex 3369	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
69	68	adantr 486	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
70	69	ad2antrr 739	. . . . . . . . 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 17419	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐿 → (𝑈 ∩ 𝐵) = (Base‘𝐼))
72	71	ad3antlr 744	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈 ∩ 𝐵) = (Base‘𝐼))
73		ineq1 4110	. . . . . . . . . . . . 13 ⊢ (𝑈 = 𝐵 → (𝑈 ∩ 𝐵) = (𝐵 ∩ 𝐵))
74		inidm 4123	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐵) = 𝐵
75	73, 74	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ (𝑈 = 𝐵 → (𝑈 ∩ 𝐵) = 𝐵)
76	75	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈 ∩ 𝐵) = 𝐵)
77	72, 76	eqtr3d 2797	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (Base‘𝐼) = 𝐵)
78	20	ad3antlr 744	. . . . . . . . . . 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 3334	. . . . . . . . . 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 3335	. . . . . . . . 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 260	. . . . . . . 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 418	. . . . . . 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 18	. . . . . 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 873	. . . . 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 215	. . . 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 282	. . 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 700	. 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 286	1 ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363   ∩ cin 4089  {csn 4584  ‘cfv 6538  (class class class)co 7420  Basecbs 17392   ↾_s cress 17413  ._rcmulr 17434  0_gc0g 17615  Rngcrng 20379  1_rcur 20412  Ringcrg 20464  Domncdomn 20949  LIdealclidl 21489

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-mulr 17447  df-sca 17449  df-vsca 17450  df-ip 17451  df-0g 17617  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-grp 19152  df-minusg 19153  df-sbg 19154  df-subg 19338  df-cmn 20001  df-abl 20002  df-mgp 20366  df-rng 20380  df-ur 20413  df-ring 20466  df-nzr 20768  df-subrg 20827  df-domn 20952  df-lmod 21142  df-lss 21212  df-sra 21453  df-rgmod 21454  df-lidl 21491

This theorem is used by:  lidldomnnring  49426