| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . 3
⊢
(Base‘𝐼) =
(Base‘𝐼) |
| 2 | | eqid 2737 |
. . 3
⊢
(.r‘𝐼) = (.r‘𝐼) |
| 3 | 1, 2 | isringrng 20284 |
. 2
⊢ (𝐼 ∈ Ring ↔ (𝐼 ∈ Rng ∧ ∃𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)((𝑥(.r‘𝐼)𝑦) = 𝑦 ∧ (𝑦(.r‘𝐼)𝑥) = 𝑦))) |
| 4 | | domnring 20707 |
. . . . 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 48149 |
. . . 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 2737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 14 | 7, 13 | ressmulr 17351 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼)) |
| 15 | 14 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . 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 2739 |
. . . . . . . . . . . . . . . . 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 2739 |
. . . . . . . . . . . . . . . . 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 3178 |
. . . . . . . . . . . . 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 21224 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈) |
| 26 | 25 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . 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 2739 |
. . . . . . . . . . . . . . . . . . 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 2954 |
. . . . . . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . 15
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 38 | | zlidlring.0 |
. . . . . . . . . . . . . . 15
⊢ 0 =
(0g‘𝑅) |
| 39 | 6, 13, 37, 38 | lidldomn1 48147 |
. . . . . . . . . . . . . 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 1373 |
. . . . . . . . . . . . 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 2827 |
. . . . . . . . . . . . . 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 2842 |
. . . . . . . . . 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 3157 |
. . . . . . . . 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 21236 |
. . . . . . . . . 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 864 |
. . . . . 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 48150 |
. . . . . . . . . 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 20251 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring →
∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)) |
| 67 | | reurex 3384 |
. . . . . . . . . . . 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 4213 |
. . . . . . . . . . . . 13
⊢ (𝑈 = 𝐵 → (𝑈 ∩ 𝐵) = (𝐵 ∩ 𝐵)) |
| 74 | | inidm 4227 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∩ 𝐵) = 𝐵 |
| 75 | 73, 74 | eqtrdi 2793 |
. . . . . . . . . . . 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 2779 |
. . . . . . . . . 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 3346 |
. . . . . . . . . 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 3347 |
. . . . . . . . 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 860 |
. . . . 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))) |