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Theorem uzlidlring 41700
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 = (𝑅s 𝑈)
zlidlring.b 𝐵 = (Base‘𝑅)
zlidlring.0 0 = (0g𝑅)
Assertion
Ref Expression
uzlidlring ((𝑅 ∈ Domn ∧ 𝑈𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))

Proof of Theorem uzlidlring
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . 3 (Base‘𝐼) = (Base‘𝐼)
2 eqid 2621 . . 3 (.r𝐼) = (.r𝐼)
31, 2isringrng 41652 . 2 (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
4 domnring 19290 . . . . 5 (𝑅 ∈ Domn → 𝑅 ∈ Ring)
54anim1i 592 . . . 4 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → (𝑅 ∈ Ring ∧ 𝑈𝐿))
6 lidlabl.l . . . . 5 𝐿 = (LIdeal‘𝑅)
7 lidlabl.i . . . . 5 𝐼 = (𝑅s 𝑈)
86, 7lidlrng 41698 . . . 4 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
95, 8syl 17 . . 3 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
10 ibar 525 . . . . . 6 (𝐼 ∈ Rng → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦))))
1110bicomd 213 . . . . 5 (𝐼 ∈ Rng → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) ↔ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
1211adantl 482 . . . 4 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) ↔ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
13 eqid 2621 . . . . . . . . . . . . . . . . . . . . . 22 (.r𝑅) = (.r𝑅)
147, 13ressmulr 16000 . . . . . . . . . . . . . . . . . . . . 21 (𝑈𝐿 → (.r𝑅) = (.r𝐼))
1514eqcomd 2627 . . . . . . . . . . . . . . . . . . . 20 (𝑈𝐿 → (.r𝐼) = (.r𝑅))
1615oveqd 6664 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → (𝑥(.r𝐼)𝑦) = (𝑥(.r𝑅)𝑦))
1716eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → ((𝑥(.r𝐼)𝑦) = 𝑦 ↔ (𝑥(.r𝑅)𝑦) = 𝑦))
1815oveqd 6664 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → (𝑦(.r𝐼)𝑥) = (𝑦(.r𝑅)𝑥))
1918eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → ((𝑦(.r𝐼)𝑥) = 𝑦 ↔ (𝑦(.r𝑅)𝑥) = 𝑦))
2017, 19anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑈𝐿 → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
2120ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
2221ad2antrr 762 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
2322ralbidv 2985 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
24 simp-4l 806 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑅 ∈ Domn)
256, 7lidlbas 41694 . . . . . . . . . . . . . . . . . . . 20 (𝑈𝐿 → (Base‘𝐼) = 𝑈)
2625eleq1d 2685 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → ((Base‘𝐼) ∈ 𝐿𝑈𝐿))
2726ibir 257 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → (Base‘𝐼) ∈ 𝐿)
2827ad3antlr 767 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ∈ 𝐿)
2925ad2antlr 763 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (Base‘𝐼) = 𝑈)
3029eqeq1d 2623 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } ↔ 𝑈 = { 0 }))
3130biimpd 219 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } → 𝑈 = { 0 }))
3231necon3bd 2807 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (¬ 𝑈 = { 0 } → (Base‘𝐼) ≠ { 0 }))
3332imp 445 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ≠ { 0 })
3428, 33jca 554 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
3534adantr 481 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
36 simpr 477 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ (Base‘𝐼))
37 eqid 2621 . . . . . . . . . . . . . . . 16 (1r𝑅) = (1r𝑅)
38 zlidlring.0 . . . . . . . . . . . . . . . 16 0 = (0g𝑅)
396, 13, 37, 38lidldomn1 41692 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Domn ∧ ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦) → 𝑥 = (1r𝑅)))
4024, 35, 36, 39syl3anc 1325 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦) → 𝑥 = (1r𝑅)))
4123, 40sylbid 230 . . . . . . . . . . . . 13 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) → 𝑥 = (1r𝑅)))
4241imp 445 . . . . . . . . . . . 12 ((((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → 𝑥 = (1r𝑅))
4325ad3antlr 767 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) = 𝑈)
4443eleq2d 2686 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) ↔ 𝑥𝑈))
4544biimpd 219 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) → 𝑥𝑈))
4645imp 445 . . . . . . . . . . . . 13 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥𝑈)
4746adantr 481 . . . . . . . . . . . 12 ((((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → 𝑥𝑈)
4842, 47eqeltrrd 2701 . . . . . . . . . . 11 ((((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → (1r𝑅) ∈ 𝑈)
4948ex 450 . . . . . . . . . 10 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) → (1r𝑅) ∈ 𝑈))
5049rexlimdva 3029 . . . . . . . . 9 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) → (1r𝑅) ∈ 𝑈))
5150impancom 456 . . . . . . . 8 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → (¬ 𝑈 = { 0 } → (1r𝑅) ∈ 𝑈))
525adantr 481 . . . . . . . . . 10 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑅 ∈ Ring ∧ 𝑈𝐿))
53 zlidlring.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
546, 53, 37lidl1el 19212 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5552, 54syl 17 . . . . . . . . 9 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5655adantr 481 . . . . . . . 8 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5751, 56sylibd 229 . . . . . . 7 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → (¬ 𝑈 = { 0 } → 𝑈 = 𝐵))
5857orrd 393 . . . . . 6 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → (𝑈 = { 0 } ∨ 𝑈 = 𝐵))
5958ex 450 . . . . 5 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) → (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
606, 7, 53, 38zlidlring 41699 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
613simprbi 480 . . . . . . . . . 10 (𝐼 ∈ Ring → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦))
6260, 61syl 17 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦))
6362ex 450 . . . . . . . 8 (𝑅 ∈ Ring → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
644, 63syl 17 . . . . . . 7 (𝑅 ∈ Domn → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
6564ad2antrr 762 . . . . . 6 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
665anim1i 592 . . . . . . 7 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng))
6753, 13ringideu 18559 . . . . . . . . . . . 12 (𝑅 ∈ Ring → ∃!𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
68 reurex 3158 . . . . . . . . . . . 12 (∃!𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦) → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
6967, 68syl 17 . . . . . . . . . . 11 (𝑅 ∈ Ring → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
7069adantr 481 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
7170ad2antrr 762 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
727, 53ressbas 15924 . . . . . . . . . . . 12 (𝑈𝐿 → (𝑈𝐵) = (Base‘𝐼))
7372ad3antlr 767 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈𝐵) = (Base‘𝐼))
74 ineq1 3805 . . . . . . . . . . . . 13 (𝑈 = 𝐵 → (𝑈𝐵) = (𝐵𝐵))
75 inidm 3820 . . . . . . . . . . . . 13 (𝐵𝐵) = 𝐵
7674, 75syl6eq 2671 . . . . . . . . . . . 12 (𝑈 = 𝐵 → (𝑈𝐵) = 𝐵)
7776adantl 482 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈𝐵) = 𝐵)
7873, 77eqtr3d 2657 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (Base‘𝐼) = 𝐵)
7920ad3antlr 767 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
8078, 79raleqbidv 3150 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
8178, 80rexeqbidv 3151 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
8271, 81mpbird 247 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦))
8382ex 450 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = 𝐵 → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
8466, 83syl 17 . . . . . 6 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = 𝐵 → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
8565, 84jaod 395 . . . . 5 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((𝑈 = { 0 } ∨ 𝑈 = 𝐵) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
8659, 85impbid 202 . . . 4 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
8712, 86bitrd 268 . . 3 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
889, 87mpdan 702 . 2 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
893, 88syl5bb 272 1 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1482  wcel 1989  wne 2793  wral 2911  wrex 2912  ∃!wreu 2913  cin 3571  {csn 4175  cfv 5886  (class class class)co 6647  Basecbs 15851  s cress 15852  .rcmulr 15936  0gc0g 16094  1rcur 18495  Ringcrg 18541  LIdealclidl 19164  Domncdomn 19274  Rngcrng 41645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-7 11081  df-8 11082  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-mulr 15949  df-sca 15951  df-vsca 15952  df-ip 15953  df-0g 16096  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-grp 17419  df-minusg 17420  df-sbg 17421  df-subg 17585  df-cmn 18189  df-abl 18190  df-mgp 18484  df-ur 18496  df-ring 18543  df-subrg 18772  df-lmod 18859  df-lss 18927  df-sra 19166  df-rgmod 19167  df-lidl 19168  df-nzr 19252  df-domn 19278  df-rng0 41646
This theorem is referenced by:  lidldomnnring  41701
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