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Theorem uzlidlring 48151
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 20284 . 2 (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦)))
4 domnring 20707 . . . . 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 48149 . . . 4 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
95, 8syl 17 . . 3 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
10 ibar 528 . . . . . 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 223 . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
147, 13ressmulr 17351 . . . . . . . . . . . . . . . . . . . 20 (𝑈𝐿 → (.r𝑅) = (.r𝐼))
1514eqcomd 2743 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → (.r𝐼) = (.r𝑅))
1615oveqd 7448 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → (𝑥(.r𝐼)𝑦) = (𝑥(.r𝑅)𝑦))
1716eqeq1d 2739 . . . . . . . . . . . . . . . . 17 (𝑈𝐿 → ((𝑥(.r𝐼)𝑦) = 𝑦 ↔ (𝑥(.r𝑅)𝑦) = 𝑦))
1815oveqd 7448 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → (𝑦(.r𝐼)𝑥) = (𝑦(.r𝑅)𝑥))
1918eqeq1d 2739 . . . . . . . . . . . . . . . . 17 (𝑈𝐿 → ((𝑦(.r𝐼)𝑥) = 𝑦 ↔ (𝑦(.r𝑅)𝑥) = 𝑦))
2017, 19anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑈𝐿 → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
2120ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
2221ad2antrr 726 . . . . . . . . . . . . . 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 3178 . . . . . . . . . . . . 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 783 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑅 ∈ Domn)
256, 7lidlbas 21224 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐿 → (Base‘𝐼) = 𝑈)
2625eleq1d 2826 . . . . . . . . . . . . . . . . . 18 (𝑈𝐿 → ((Base‘𝐼) ∈ 𝐿𝑈𝐿))
2726ibir 268 . . . . . . . . . . . . . . . . 17 (𝑈𝐿 → (Base‘𝐼) ∈ 𝐿)
2827ad3antlr 731 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ∈ 𝐿)
2925ad2antlr 727 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (Base‘𝐼) = 𝑈)
3029eqeq1d 2739 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } ↔ 𝑈 = { 0 }))
3130biimpd 229 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((Base‘𝐼) = { 0 } → 𝑈 = { 0 }))
3231necon3bd 2954 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (¬ 𝑈 = { 0 } → (Base‘𝐼) ≠ { 0 }))
3332imp 406 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) ≠ { 0 })
3428, 33jca 511 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
3534adantr 480 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → ((Base‘𝐼) ∈ 𝐿 ∧ (Base‘𝐼) ≠ { 0 }))
36 simpr 484 . . . . . . . . . . . . . 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 48147 . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 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 240 . . . . . . . . . . . 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 406 . . . . . . . . . . 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 731 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (Base‘𝐼) = 𝑈)
4443eleq2d 2827 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) ↔ 𝑥𝑈))
4544biimpd 229 . . . . . . . . . . . . 13 ((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) → (𝑥 ∈ (Base‘𝐼) → 𝑥𝑈))
4645imp 406 . . . . . . . . . . . 12 (((((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ ¬ 𝑈 = { 0 }) ∧ 𝑥 ∈ (Base‘𝐼)) → 𝑥𝑈)
4746adantr 480 . . . . . . . . . . 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 2842 . . . . . . . . . 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 3157 . . . . . . . . 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 451 . . . . . . . 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 480 . . . . . . . . . 10 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → (𝑅 ∈ Ring ∧ 𝑈𝐿))
52 zlidlring.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
536, 52, 37lidl1el 21236 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5451, 53syl 17 . . . . . . . . 9 (((𝑅 ∈ Domn ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) → ((1r𝑅) ∈ 𝑈𝑈 = 𝐵))
5554adantr 480 . . . . . . . 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 239 . . . . . . 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 864 . . . . . 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 412 . . . . 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 48150 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
603simprbi 496 . . . . . . . . . 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 412 . . . . . . . 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 726 . . . . . 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 20251 . . . . . . . . . . . 12 (𝑅 ∈ Ring → ∃!𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
67 reurex 3384 . . . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑈𝐿) → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
7069ad2antrr 726 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → ∃𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))
717, 52ressbas 17280 . . . . . . . . . . . 12 (𝑈𝐿 → (𝑈𝐵) = (Base‘𝐼))
7271ad3antlr 731 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → (𝑈𝐵) = (Base‘𝐼))
73 ineq1 4213 . . . . . . . . . . . . 13 (𝑈 = 𝐵 → (𝑈𝐵) = (𝐵𝐵))
74 inidm 4227 . . . . . . . . . . . . 13 (𝐵𝐵) = 𝐵
7573, 74eqtrdi 2793 . . . . . . . . . . . 12 (𝑈 = 𝐵 → (𝑈𝐵) = 𝐵)
7675adantl 481 . . . . . . . . . . 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 731 . . . . . . . . . . 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 3346 . . . . . . . . . 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 3347 . . . . . . . . 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 257 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑈𝐿) ∧ 𝐼 ∈ Rng) ∧ 𝑈 = 𝐵) → ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r𝐼)𝑦) = 𝑦 ∧ (𝑦(.r𝐼)𝑥) = 𝑦))
8281ex 412 . . . . . . 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 860 . . . . 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 212 . . . 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 279 . . 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 687 . 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 283 1 ((𝑅 ∈ Domn ∧ 𝑈𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  ∃!wreu 3378  cin 3950  {csn 4626  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  .rcmulr 17298  0gc0g 17484  Rngcrng 20149  1rcur 20178  Ringcrg 20230  Domncdomn 20692  LIdealclidl 21216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-nzr 20513  df-subrg 20570  df-domn 20695  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218
This theorem is referenced by:  lidldomnnring  48152
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