| Step | Hyp | Ref
| Expression |
| 1 | | subrdom.1 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ Domn) |
| 2 | | domnnzr 20706 |
. . . 4
⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) |
| 3 | 1, 2 | syl 17 |
. . 3
⊢ (𝜑 → 𝑅 ∈ NzRing) |
| 4 | | subrdom.2 |
. . 3
⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) |
| 5 | | eqid 2737 |
. . . 4
⊢ (𝑅 ↾s 𝑆) = (𝑅 ↾s 𝑆) |
| 6 | 5 | subrgnzr 20594 |
. . 3
⊢ ((𝑅 ∈ NzRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → (𝑅 ↾s 𝑆) ∈ NzRing) |
| 7 | 3, 4, 6 | syl2anc 584 |
. 2
⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ NzRing) |
| 8 | 1 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑅 ∈ Domn) |
| 9 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘𝑅) =
(Base‘𝑅) |
| 10 | 9 | subrgss 20572 |
. . . . . . . . . 10
⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ (Base‘𝑅)) |
| 11 | 4, 10 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑅)) |
| 12 | 11 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑆 ⊆ (Base‘𝑅)) |
| 13 | | simpllr 776 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) |
| 14 | 5, 9 | ressbas2 17283 |
. . . . . . . . . . 11
⊢ (𝑆 ⊆ (Base‘𝑅) → 𝑆 = (Base‘(𝑅 ↾s 𝑆))) |
| 15 | 11, 14 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 = (Base‘(𝑅 ↾s 𝑆))) |
| 16 | 15 | ad3antrrr 730 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑆 = (Base‘(𝑅 ↾s 𝑆))) |
| 17 | 13, 16 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑥 ∈ 𝑆) |
| 18 | 12, 17 | sseldd 3984 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑥 ∈ (Base‘𝑅)) |
| 19 | | simplr 769 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) |
| 20 | 19, 16 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑦 ∈ 𝑆) |
| 21 | 12, 20 | sseldd 3984 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → 𝑦 ∈ (Base‘𝑅)) |
| 22 | | simpr 484 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) |
| 23 | 4 | elexd 3504 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑆 ∈ V) |
| 24 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 25 | 5, 24 | ressmulr 17351 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ V →
(.r‘𝑅) =
(.r‘(𝑅
↾s 𝑆))) |
| 26 | 23, 25 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (.r‘𝑅) = (.r‘(𝑅 ↾s 𝑆))) |
| 27 | 26 | oveqd 7448 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦)) |
| 28 | 27 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦)) |
| 29 | | subrgrcl 20576 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) |
| 30 | | ringmnd 20240 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) |
| 31 | 4, 29, 30 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈ Mnd) |
| 32 | | subrgsubg 20577 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ∈ (SubGrp‘𝑅)) |
| 33 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(0g‘𝑅) = (0g‘𝑅) |
| 34 | 33 | subg0cl 19152 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ (SubGrp‘𝑅) →
(0g‘𝑅)
∈ 𝑆) |
| 35 | 4, 32, 34 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (0g‘𝑅) ∈ 𝑆) |
| 36 | 5, 9, 33 | ress0g 18775 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Mnd ∧
(0g‘𝑅)
∈ 𝑆 ∧ 𝑆 ⊆ (Base‘𝑅)) →
(0g‘𝑅) =
(0g‘(𝑅
↾s 𝑆))) |
| 37 | 31, 35, 11, 36 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝑆))) |
| 38 | 37 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝑆))) |
| 39 | 22, 28, 38 | 3eqtr4d 2787 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅)) |
| 40 | 9, 24, 33 | domneq0 20708 |
. . . . . . . 8
⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) ↔ (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
| 41 | 40 | biimpa 476 |
. . . . . . 7
⊢ (((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅)) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) |
| 42 | 8, 18, 21, 39, 41 | syl31anc 1375 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) |
| 43 | 38 | eqeq2d 2748 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → (𝑥 = (0g‘𝑅) ↔ 𝑥 = (0g‘(𝑅 ↾s 𝑆)))) |
| 44 | 38 | eqeq2d 2748 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → (𝑦 = (0g‘𝑅) ↔ 𝑦 = (0g‘(𝑅 ↾s 𝑆)))) |
| 45 | 43, 44 | orbi12d 919 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑥 = (0g‘(𝑅 ↾s 𝑆)) ∨ 𝑦 = (0g‘(𝑅 ↾s 𝑆))))) |
| 46 | 42, 45 | mpbid 232 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ (𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆))) → (𝑥 = (0g‘(𝑅 ↾s 𝑆)) ∨ 𝑦 = (0g‘(𝑅 ↾s 𝑆)))) |
| 47 | 46 | ex 412 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))) → ((𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆)) → (𝑥 = (0g‘(𝑅 ↾s 𝑆)) ∨ 𝑦 = (0g‘(𝑅 ↾s 𝑆))))) |
| 48 | 47 | anasss 466 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝑅 ↾s 𝑆)) ∧ 𝑦 ∈ (Base‘(𝑅 ↾s 𝑆)))) → ((𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆)) → (𝑥 = (0g‘(𝑅 ↾s 𝑆)) ∨ 𝑦 = (0g‘(𝑅 ↾s 𝑆))))) |
| 49 | 48 | ralrimivva 3202 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ (Base‘(𝑅 ↾s 𝑆))∀𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))((𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆)) → (𝑥 = (0g‘(𝑅 ↾s 𝑆)) ∨ 𝑦 = (0g‘(𝑅 ↾s 𝑆))))) |
| 50 | | eqid 2737 |
. . 3
⊢
(Base‘(𝑅
↾s 𝑆)) =
(Base‘(𝑅
↾s 𝑆)) |
| 51 | | eqid 2737 |
. . 3
⊢
(.r‘(𝑅 ↾s 𝑆)) = (.r‘(𝑅 ↾s 𝑆)) |
| 52 | | eqid 2737 |
. . 3
⊢
(0g‘(𝑅 ↾s 𝑆)) = (0g‘(𝑅 ↾s 𝑆)) |
| 53 | 50, 51, 52 | isdomn 20705 |
. 2
⊢ ((𝑅 ↾s 𝑆) ∈ Domn ↔ ((𝑅 ↾s 𝑆) ∈ NzRing ∧
∀𝑥 ∈
(Base‘(𝑅
↾s 𝑆))∀𝑦 ∈ (Base‘(𝑅 ↾s 𝑆))((𝑥(.r‘(𝑅 ↾s 𝑆))𝑦) = (0g‘(𝑅 ↾s 𝑆)) → (𝑥 = (0g‘(𝑅 ↾s 𝑆)) ∨ 𝑦 = (0g‘(𝑅 ↾s 𝑆)))))) |
| 54 | 7, 49, 53 | sylanbrc 583 |
1
⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ Domn) |