| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | isdomn2.b | . . 3
⊢ 𝐵 = (Base‘𝑅) | 
| 2 |  | eqid 2736 | . . 3
⊢
(.r‘𝑅) = (.r‘𝑅) | 
| 3 |  | isdomn2.z | . . 3
⊢  0 =
(0g‘𝑅) | 
| 4 | 1, 2, 3 | isdomn 20706 | . 2
⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧
∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))) | 
| 5 |  | dfss3 3971 | . . . 4
⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸) | 
| 6 |  | isdomn2.t | . . . . . . . . 9
⊢ 𝐸 = (RLReg‘𝑅) | 
| 7 | 6, 1, 2, 3 | isrrg 20699 | . . . . . . . 8
⊢ (𝑥 ∈ 𝐸 ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 8 | 7 | baib 535 | . . . . . . 7
⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐸 ↔ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 9 | 8 | imbi2d 340 | . . . . . 6
⊢ (𝑥 ∈ 𝐵 → ((𝑥 ≠ 0 → 𝑥 ∈ 𝐸) ↔ (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))) | 
| 10 | 9 | ralbiia 3090 | . . . . 5
⊢
(∀𝑥 ∈
𝐵 (𝑥 ≠ 0 → 𝑥 ∈ 𝐸) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 11 |  | eldifsn 4785 | . . . . . . . 8
⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) | 
| 12 | 11 | imbi1i 349 | . . . . . . 7
⊢ ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐸) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝐸)) | 
| 13 |  | impexp 450 | . . . . . . 7
⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝐸) ↔ (𝑥 ∈ 𝐵 → (𝑥 ≠ 0 → 𝑥 ∈ 𝐸))) | 
| 14 | 12, 13 | bitri 275 | . . . . . 6
⊢ ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐸) ↔ (𝑥 ∈ 𝐵 → (𝑥 ≠ 0 → 𝑥 ∈ 𝐸))) | 
| 15 | 14 | ralbii2 3088 | . . . . 5
⊢
(∀𝑥 ∈
(𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → 𝑥 ∈ 𝐸)) | 
| 16 |  | con34b 316 | . . . . . . . . 9
⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (¬ (𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 )) | 
| 17 |  | impexp 450 | . . . . . . . . . 10
⊢ (((¬
𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ))) | 
| 18 |  | ioran 985 | . . . . . . . . . . 11
⊢ (¬
(𝑥 = 0 ∨ 𝑦 = 0 ) ↔ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 )) | 
| 19 | 18 | imbi1i 349 | . . . . . . . . . 10
⊢ ((¬
(𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ ((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 )) | 
| 20 |  | df-ne 2940 | . . . . . . . . . . 11
⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0 ) | 
| 21 |  | con34b 316 | . . . . . . . . . . 11
⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ) ↔ (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 )) | 
| 22 | 20, 21 | imbi12i 350 | . . . . . . . . . 10
⊢ ((𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ))) | 
| 23 | 17, 19, 22 | 3bitr4i 303 | . . . . . . . . 9
⊢ ((¬
(𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 24 | 16, 23 | bitri 275 | . . . . . . . 8
⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 25 | 24 | ralbii 3092 | . . . . . . 7
⊢
(∀𝑦 ∈
𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 26 |  | r19.21v 3179 | . . . . . . 7
⊢
(∀𝑦 ∈
𝐵 (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )) ↔ (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 27 | 25, 26 | bitri 275 | . . . . . 6
⊢
(∀𝑦 ∈
𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 28 | 27 | ralbii 3092 | . . . . 5
⊢
(∀𝑥 ∈
𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))) | 
| 29 | 10, 15, 28 | 3bitr4i 303 | . . . 4
⊢
(∀𝑥 ∈
(𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) | 
| 30 | 5, 29 | bitr2i 276 | . . 3
⊢
(∀𝑥 ∈
𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝐵 ∖ { 0 }) ⊆ 𝐸) | 
| 31 | 30 | anbi2i 623 | . 2
⊢ ((𝑅 ∈ NzRing ∧
∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸)) | 
| 32 | 4, 31 | bitri 275 | 1
⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸)) |