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Theorem uzlidlring 42254
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 2651 . . 3 (Base‘𝐼) = (Base‘𝐼)
2 eqid 2651 . . 3 (.r𝐼) = (.r𝐼)
31, 2isringrng 42206 . 2 (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
4 domnring 19344 . . . . 5 (𝑅 ∈ Domn → 𝑅 ∈ Ring)
54anim1i 591 . . . 4 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → (𝑅 ∈ Ring ∧ 𝑈𝐿))
6 lidlabl.l . . . . 5 𝐿 = (LIdeal‘𝑅)
7 lidlabl.i . . . . 5 𝐼 = (𝑅s 𝑈)
86, 7lidlrng 42252 . . . 4 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
95, 8syl 17 . . 3 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
10 ibar 524 . . . . . 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 481 . . . 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 2651 . . . . . . . . . . . . . . . . . . . . . 22 (.r𝑅) = (.r𝑅)
147, 13ressmulr 16053 . . . . . . . . . . . . . . . . . . . . 21 (𝑈𝐿 → (.r𝑅) = (.r𝐼))
1514eqcomd 2657 . . . . . . . . . . . . . . . . . . . 20 (𝑈𝐿 → (.r𝐼) = (.r𝑅))
1615oveqd 6707 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → (𝑥(.r𝐼)𝑦) = (𝑥(.r𝑅)𝑦))
1716eqeq1d 2653 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → ((𝑥(.r𝐼)𝑦) = 𝑦 ↔ (𝑥(.r𝑅)𝑦) = 𝑦))
1815oveqd 6707 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → (𝑦(.r𝐼)𝑥) = (𝑦(.r𝑅)𝑥))
1918eqeq1d 2653 . . . . . . . . . . . . . . . . . 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 3015 . . . . . . . . . . . . . 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 823 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑅 ∈ Domn)
256, 7lidlbas 42248 . . . . . . . . . . . . . . . . . . . 20 (𝑈𝐿 → (Base‘𝐼) = 𝑈)
2625eleq1d 2715 . . . . . . . . . . . . . . . . . . 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 2653 . . . . . . . . . . . . . . . . . . . 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 2837 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (¬ 𝑈 = { 0 } → (Base‘𝐼) ≠ { 0 }))
3332imp 444 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ≠ { 0 })
3428, 33jca 553 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
3534adantr 480 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
36 simpr 476 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ (Base‘𝐼))
37 eqid 2651 . . . . . . . . . . . . . . . 16 (1r𝑅) = (1r𝑅)
38 zlidlring.0 . . . . . . . . . . . . . . . 16 0 = (0g𝑅)
396, 13, 37, 38lidldomn1 42246 . . . . . . . . . . . . . . 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 1366 . . . . . . . . . . . . . 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 444 . . . . . . . . . . . 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 2716 . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . 13 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥𝑈)
4746adantr 480 . . . . . . . . . . . 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 2731 . . . . . . . . . . 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 449 . . . . . . . . . 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 3060 . . . . . . . . 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 455 . . . . . . . 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 480 . . . . . . . . . 10 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑅 ∈ Ring ∧ 𝑈𝐿))
53 zlidlring.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
546, 53, 37lidl1el 19266 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5552, 54syl 17 . . . . . . . . 9 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5655adantr 480 . . . . . . . 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 392 . . . . . 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 449 . . . . 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 42253 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
613simprbi 479 . . . . . . . . . 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 449 . . . . . . . 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 591 . . . . . . 7 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng))
6753, 13ringideu 18611 . . . . . . . . . . . 12 (𝑅 ∈ Ring → ∃!𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
68 reurex 3190 . . . . . . . . . . . 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 480 . . . . . . . . . 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 15977 . . . . . . . . . . . 12 (𝑈𝐿 → (𝑈𝐵) = (Base‘𝐼))
7372ad3antlr 767 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈𝐵) = (Base‘𝐼))
74 ineq1 3840 . . . . . . . . . . . . 13 (𝑈 = 𝐵 → (𝑈𝐵) = (𝐵𝐵))
75 inidm 3855 . . . . . . . . . . . . 13 (𝐵𝐵) = 𝐵
7674, 75syl6eq 2701 . . . . . . . . . . . 12 (𝑈 = 𝐵 → (𝑈𝐵) = 𝐵)
7776adantl 481 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈𝐵) = 𝐵)
7873, 77eqtr3d 2687 . . . . . . . . . 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 3182 . . . . . . . . . 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 3183 . . . . . . . . 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 449 . . . . . . 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 394 . . . . 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 703 . 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 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  cin 3606  {csn 4210  cfv 5926  (class class class)co 6690  Basecbs 15904  s cress 15905  .rcmulr 15989  0gc0g 16147  1rcur 18547  Ringcrg 18593  LIdealclidl 19218  Domncdomn 19328  Rngcrng 42199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-lidl 19222  df-nzr 19306  df-domn 19332  df-rng0 42200
This theorem is referenced by:  lidldomnnring  42255
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