Theorem uzlidlring 48079

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 2735	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
2		eqid 2735	. . 3 ⊢ (.r‘𝐼) = (.r‘𝐼)
3	1, 2	isringrng 20301	. 2 ⊢ (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦)))
4		domnring 20724	. . . . 5 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)
5	4	anim1i 615	. . . 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 48077	. . . 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 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (.r‘𝑅) = (.r‘𝑅)
14	7, 13	ressmulr 17353	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼))
15	14	eqcomd 2741	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝐼) = (.r‘𝑅))
16	15	oveqd 7448	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → (𝑥(.r‘𝐼)𝑦) = (𝑥(.r‘𝑅)𝑦))
17	16	eqeq1d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → ((𝑥(.r‘𝐼)𝑦) = 𝑦 ↔ (𝑥(.r‘𝑅)𝑦) = 𝑦))
18	15	oveqd 7448	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → (𝑦(.r‘𝐼)𝑥) = (𝑦(.r‘𝑅)𝑥))
19	18	eqeq1d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → ((𝑦(.r‘𝐼)𝑥) = 𝑦 ↔ (𝑦(.r‘𝑅)𝑥) = 𝑦))
20	17, 19	anbi12d 632	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . 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 21242	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
26	25	eleq1d 2824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) ∈ 𝐿 ↔ 𝑈 ∈ 𝐿))
27	26	ibir 268	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿)
28	27	ad3antlr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ∈ 𝐿)
29	25	ad2antlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → (Base‘𝐼) = 𝑈)
30	29	eqeq1d 2737	. . . . . . . . . . . . . . . . . . 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 2952	. . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑅) = (1r‘𝑅)
38		zlidlring.0	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝑅)
39	6, 13, 37, 38	lidldomn1 48075	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) = 𝑈)
44	43	eleq2d 2825	. . . . . . . . . . . . . 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 2840	. . . . . . . . . 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 3155	. . . . . . . . 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 21254	. . . . . . . . . 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 863	. . . . . 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 48078	. . . . . . . . . 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 726	. . . . . 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 615	. . . . . . 7 ⊢ (((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) → ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng))
66	52, 13	ringideu 20272	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))
67		reurex 3382	. . . . . . . . . . . 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 726	. . . . . . . . 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 17280	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐿 → (𝑈 ∩ 𝐵) = (Base‘𝐼))
72	71	ad3antlr 731	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈 ∩ 𝐵) = (Base‘𝐼))
73		ineq1 4221	. . . . . . . . . . . . 13 ⊢ (𝑈 = 𝐵 → (𝑈 ∩ 𝐵) = (𝐵 ∩ 𝐵))
74		inidm 4235	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐵) = 𝐵
75	73, 74	eqtrdi 2791	. . . . . . . . . . . 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 2777	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (Base‘𝐼) = 𝐵)
78	20	ad3antlr 731	. . . . . . . . . . 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 3344	. . . . . . . . . 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 3345	. . . . . . . . 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 859	. . . . 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 687	. 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 847   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  ∃!wreu 3376   ∩ cin 3962  {csn 4631  ‘cfv 6563  (class class class)co 7431  Basecbs 17245   ↾_s cress 17274  ._rcmulr 17299  0_gc0g 17486  Rngcrng 20170  1_rcur 20199  Ringcrg 20251  Domncdomn 20709  LIdealclidl 21234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-nzr 20530  df-subrg 20587  df-domn 20712  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236

This theorem is referenced by:  lidldomnnring  48080