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Theorem uzlidlring 45746
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 2737 . . 3 (Base‘𝐼) = (Base‘𝐼)
2 eqid 2737 . . 3 (.r𝐼) = (.r𝐼)
31, 2isringrng 45698 . 2 (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
4 domnring 20638 . . . . 5 (𝑅 ∈ Domn → 𝑅 ∈ Ring)
54anim1i 615 . . . 4 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → (𝑅 ∈ Ring ∧ 𝑈𝐿))
6 lidlabl.l . . . . 5 𝐿 = (LIdeal‘𝑅)
7 lidlabl.i . . . . 5 𝐼 = (𝑅s 𝑈)
86, 7lidlrng 45744 . . . 4 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
95, 8syl 17 . . 3 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
10 ibar 529 . . . . . 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 222 . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
147, 13ressmulr 17084 . . . . . . . . . . . . . . . . . . . 20 (𝑈𝐿 → (.r𝑅) = (.r𝐼))
1514eqcomd 2743 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → (.r𝐼) = (.r𝑅))
1615oveqd 7330 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → (𝑥(.r𝐼)𝑦) = (𝑥(.r𝑅)𝑦))
1716eqeq1d 2739 . . . . . . . . . . . . . . . . 17 (𝑈𝐿 → ((𝑥(.r𝐼)𝑦) = 𝑦 ↔ (𝑥(.r𝑅)𝑦) = 𝑦))
1815oveqd 7330 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → (𝑦(.r𝐼)𝑥) = (𝑦(.r𝑅)𝑥))
1918eqeq1d 2739 . . . . . . . . . . . . . . . . 17 (𝑈𝐿 → ((𝑦(.r𝐼)𝑥) = 𝑦 ↔ (𝑦(.r𝑅)𝑥) = 𝑦))
2017, 19anbi12d 631 . . . . . . . . . . . . . . . 16 (𝑈𝐿 → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
2120ad2antlr 724 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
2221ad2antrr 723 . . . . . . . . . . . . . 14 (((((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 3171 . . . . . . . . . . . . 13 (((((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 780 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑅 ∈ Domn)
256, 7lidlbas 45740 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → (Base‘𝐼) = 𝑈)
2625eleq1d 2822 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → ((Base‘𝐼) ∈ 𝐿𝑈𝐿))
2726ibir 267 . . . . . . . . . . . . . . . . 17 (𝑈𝐿 → (Base‘𝐼) ∈ 𝐿)
2827ad3antlr 728 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ∈ 𝐿)
2925ad2antlr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (Base‘𝐼) = 𝑈)
3029eqeq1d 2739 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } ↔ 𝑈 = { 0 }))
3130biimpd 228 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } → 𝑈 = { 0 }))
3231necon3bd 2955 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (¬ 𝑈 = { 0 } → (Base‘𝐼) ≠ { 0 }))
3332imp 407 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ≠ { 0 })
3428, 33jca 512 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
3534adantr 481 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
36 simpr 485 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥 ∈ (Base‘𝐼))
37 eqid 2737 . . . . . . . . . . . . . . 15 (1r𝑅) = (1r𝑅)
38 zlidlring.0 . . . . . . . . . . . . . . 15 0 = (0g𝑅)
396, 13, 37, 38lidldomn1 45738 . . . . . . . . . . . . . 14 ((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 1370 . . . . . . . . . . . . 13 (((((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 239 . . . . . . . . . . . 12 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → (∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) → 𝑥 = (1r𝑅)))
4241imp 407 . . . . . . . . . . 11 ((((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → 𝑥 = (1r𝑅))
4325ad3antlr 728 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) = 𝑈)
4443eleq2d 2823 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) ↔ 𝑥𝑈))
4544biimpd 228 . . . . . . . . . . . . 13 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) → 𝑥𝑈))
4645imp 407 . . . . . . . . . . . 12 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥𝑈)
4746adantr 481 . . . . . . . . . . 11 ((((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → 𝑥𝑈)
4842, 47eqeltrrd 2839 . . . . . . . . . 10 ((((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) ∧ ∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → (1r𝑅) ∈ 𝑈)
4948rexlimdva2 3151 . . . . . . . . 9 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) → (1r𝑅) ∈ 𝑈))
5049impancom 452 . . . . . . . 8 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → (¬ 𝑈 = { 0 } → (1r𝑅) ∈ 𝑈))
515adantr 481 . . . . . . . . . 10 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑅 ∈ Ring ∧ 𝑈𝐿))
52 zlidlring.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
536, 52, 37lidl1el 20560 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5451, 53syl 17 . . . . . . . . 9 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5554adantr 481 . . . . . . . 8 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5650, 55sylibd 238 . . . . . . 7 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → (¬ 𝑈 = { 0 } → 𝑈 = 𝐵))
5756orrd 860 . . . . . 6 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) → (𝑈 = { 0 } ∨ 𝑈 = 𝐵))
5857ex 413 . . . . 5 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) → (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
596, 7, 52, 38zlidlring 45745 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
603simprbi 497 . . . . . . . . . 10 (𝐼 ∈ Ring → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦))
6159, 60syl 17 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦))
6261ex 413 . . . . . . . 8 (𝑅 ∈ Ring → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
634, 62syl 17 . . . . . . 7 (𝑅 ∈ Domn → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
6463ad2antrr 723 . . . . . 6 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = { 0 } → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
655anim1i 615 . . . . . . 7 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng))
6652, 13ringideu 19871 . . . . . . . . . . . 12 (𝑅 ∈ Ring → ∃!𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
67 reurex 3354 . . . . . . . . . . . 12 (∃!𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦) → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
6866, 67syl 17 . . . . . . . . . . 11 (𝑅 ∈ Ring → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
6968adantr 481 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
7069ad2antrr 723 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
717, 52ressbas 17014 . . . . . . . . . . . 12 (𝑈𝐿 → (𝑈𝐵) = (Base‘𝐼))
7271ad3antlr 728 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈𝐵) = (Base‘𝐼))
73 ineq1 4149 . . . . . . . . . . . . 13 (𝑈 = 𝐵 → (𝑈𝐵) = (𝐵𝐵))
74 inidm 4162 . . . . . . . . . . . . 13 (𝐵𝐵) = 𝐵
7573, 74eqtrdi 2793 . . . . . . . . . . . 12 (𝑈 = 𝐵 → (𝑈𝐵) = 𝐵)
7675adantl 482 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈𝐵) = 𝐵)
7772, 76eqtr3d 2779 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (Base‘𝐼) = 𝐵)
7820ad3antlr 728 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
7977, 78raleqbidv 3316 . . . . . . . . . 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)))
8077, 79rexeqbidv 3317 . . . . . . . . 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)))
8170, 80mpbird 256 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦))
8281ex 413 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = 𝐵 → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
8365, 82syl 17 . . . . . 6 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑈 = 𝐵 → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
8464, 83jaod 856 . . . . 5 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((𝑈 = { 0 } ∨ 𝑈 = 𝐵) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
8558, 84impbid 211 . . . 4 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
8612, 85bitrd 278 . . 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)))
879, 86mpdan 684 . 2 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → ((𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)) ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
883, 87bitrid 282 1 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1540  wcel 2105  wne 2941  wral 3062  wrex 3071  ∃!wreu 3348  cin 3895  {csn 4569  cfv 6463  (class class class)co 7313  Basecbs 16979  s cress 17008  .rcmulr 17030  0gc0g 17217  1rcur 19804  Ringcrg 19850  LIdealclidl 20503  Domncdomn 20622  Rngcrng 45691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626  ax-cnex 10997  ax-resscn 10998  ax-1cn 10999  ax-icn 11000  ax-addcl 11001  ax-addrcl 11002  ax-mulcl 11003  ax-mulrcl 11004  ax-mulcom 11005  ax-addass 11006  ax-mulass 11007  ax-distr 11008  ax-i2m1 11009  ax-1ne0 11010  ax-1rid 11011  ax-rnegex 11012  ax-rrecex 11013  ax-cnre 11014  ax-pre-lttri 11015  ax-pre-lttrn 11016  ax-pre-ltadd 11017  ax-pre-mulgt0 11018
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-pred 6222  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-riota 7270  df-ov 7316  df-oprab 7317  df-mpo 7318  df-om 7756  df-1st 7874  df-2nd 7875  df-frecs 8142  df-wrecs 8173  df-recs 8247  df-rdg 8286  df-er 8544  df-en 8780  df-dom 8781  df-sdom 8782  df-pnf 11081  df-mnf 11082  df-xr 11083  df-ltxr 11084  df-le 11085  df-sub 11277  df-neg 11278  df-nn 12044  df-2 12106  df-3 12107  df-4 12108  df-5 12109  df-6 12110  df-7 12111  df-8 12112  df-sets 16932  df-slot 16950  df-ndx 16962  df-base 16980  df-ress 17009  df-plusg 17042  df-mulr 17043  df-sca 17045  df-vsca 17046  df-ip 17047  df-0g 17219  df-mgm 18393  df-sgrp 18442  df-mnd 18453  df-grp 18647  df-minusg 18648  df-sbg 18649  df-subg 18819  df-cmn 19455  df-abl 19456  df-mgp 19788  df-ur 19805  df-ring 19852  df-subrg 20093  df-lmod 20196  df-lss 20265  df-sra 20505  df-rgmod 20506  df-lidl 20507  df-nzr 20600  df-domn 20626  df-rng0 45692
This theorem is referenced by:  lidldomnnring  45747
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