Step | Hyp | Ref
| Expression |
1 | | 3simpc 998 |
. . 3
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) |
2 | | domneq0.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
3 | | domneq0.t |
. . . . . 6
⊢ · =
(.r‘𝑅) |
4 | | domneq0.z |
. . . . . 6
⊢ 0 =
(0g‘𝑅) |
5 | 2, 3, 4 | isdomn 13743 |
. . . . 5
⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧
∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))) |
6 | 5 | simprbi 275 |
. . . 4
⊢ (𝑅 ∈ Domn →
∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) |
7 | 6 | 3ad2ant1 1020 |
. . 3
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) |
8 | | oveq1 5917 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦)) |
9 | 8 | eqeq1d 2202 |
. . . . 5
⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 )) |
10 | | eqeq1 2200 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0 )) |
11 | 10 | orbi1d 792 |
. . . . 5
⊢ (𝑥 = 𝑋 → ((𝑥 = 0 ∨ 𝑦 = 0 ) ↔ (𝑋 = 0 ∨ 𝑦 = 0 ))) |
12 | 9, 11 | imbi12d 234 |
. . . 4
⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ ((𝑋 · 𝑦) = 0 → (𝑋 = 0 ∨ 𝑦 = 0 )))) |
13 | | oveq2 5918 |
. . . . . 6
⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) |
14 | 13 | eqeq1d 2202 |
. . . . 5
⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 )) |
15 | | eqeq1 2200 |
. . . . . 6
⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 )) |
16 | 15 | orbi2d 791 |
. . . . 5
⊢ (𝑦 = 𝑌 → ((𝑋 = 0 ∨ 𝑦 = 0 ) ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) |
17 | 14, 16 | imbi12d 234 |
. . . 4
⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → (𝑋 = 0 ∨ 𝑦 = 0 )) ↔ ((𝑋 · 𝑌) = 0 → (𝑋 = 0 ∨ 𝑌 = 0 )))) |
18 | 12, 17 | rspc2va 2878 |
. . 3
⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) → ((𝑋 · 𝑌) = 0 → (𝑋 = 0 ∨ 𝑌 = 0 ))) |
19 | 1, 7, 18 | syl2anc 411 |
. 2
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → (𝑋 = 0 ∨ 𝑌 = 0 ))) |
20 | | domnring 13745 |
. . . . . 6
⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) |
21 | 20 | 3ad2ant1 1020 |
. . . . 5
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ Ring) |
22 | | simp3 1001 |
. . . . 5
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) |
23 | 2, 3, 4 | ringlz 13517 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → ( 0 · 𝑌) = 0 ) |
24 | 21, 22, 23 | syl2anc 411 |
. . . 4
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 · 𝑌) = 0 ) |
25 | | oveq1 5917 |
. . . . 5
⊢ (𝑋 = 0 → (𝑋 · 𝑌) = ( 0 · 𝑌)) |
26 | 25 | eqeq1d 2202 |
. . . 4
⊢ (𝑋 = 0 → ((𝑋 · 𝑌) = 0 ↔ ( 0 · 𝑌) = 0 )) |
27 | 24, 26 | syl5ibrcom 157 |
. . 3
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 0 → (𝑋 · 𝑌) = 0 )) |
28 | | simp2 1000 |
. . . . 5
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
29 | 2, 3, 4 | ringrz 13518 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
30 | 21, 28, 29 | syl2anc 411 |
. . . 4
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
31 | | oveq2 5918 |
. . . . 5
⊢ (𝑌 = 0 → (𝑋 · 𝑌) = (𝑋 · 0 )) |
32 | 31 | eqeq1d 2202 |
. . . 4
⊢ (𝑌 = 0 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 · 0 ) = 0 )) |
33 | 30, 32 | syl5ibrcom 157 |
. . 3
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = 0 → (𝑋 · 𝑌) = 0 )) |
34 | 27, 33 | jaod 718 |
. 2
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 = 0 ∨ 𝑌 = 0 ) → (𝑋 · 𝑌) = 0 )) |
35 | 19, 34 | impbid 129 |
1
⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) |