Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
2 |
1
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ ) |
3 |
2
|
mul02d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → ( 0 · 𝐵 ) = 0 ) |
4 |
3
|
eqeq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 · 𝐵 ) = ( 0 · 𝐵 ) ↔ ( 𝐴 · 𝐵 ) = 0 ) ) |
5 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
6 |
5
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℂ ) |
7 |
|
0cnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → 0 ∈ ℂ ) |
8 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → 𝐵 ≠ 0 ) |
9 |
6 7 2 8
|
mulcan2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 · 𝐵 ) = ( 0 · 𝐵 ) ↔ 𝐴 = 0 ) ) |
10 |
4 9
|
bitr3d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 · 𝐵 ) = 0 ↔ 𝐴 = 0 ) ) |
11 |
10
|
biimpd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 · 𝐵 ) = 0 → 𝐴 = 0 ) ) |
12 |
11
|
impancom |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐴 · 𝐵 ) = 0 ) → ( 𝐵 ≠ 0 → 𝐴 = 0 ) ) |
13 |
12
|
necon1bd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐴 · 𝐵 ) = 0 ) → ( ¬ 𝐴 = 0 → 𝐵 = 0 ) ) |
14 |
13
|
orrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐴 · 𝐵 ) = 0 ) → ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) |
15 |
14
|
ex |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) = 0 → ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) ) |
16 |
1
|
mul02d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 0 · 𝐵 ) = 0 ) |
17 |
|
oveq1 |
⊢ ( 𝐴 = 0 → ( 𝐴 · 𝐵 ) = ( 0 · 𝐵 ) ) |
18 |
17
|
eqeq1d |
⊢ ( 𝐴 = 0 → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 0 · 𝐵 ) = 0 ) ) |
19 |
16 18
|
syl5ibrcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 = 0 → ( 𝐴 · 𝐵 ) = 0 ) ) |
20 |
5
|
mul01d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 0 ) = 0 ) |
21 |
|
oveq2 |
⊢ ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐴 · 0 ) ) |
22 |
21
|
eqeq1d |
⊢ ( 𝐵 = 0 → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 · 0 ) = 0 ) ) |
23 |
20 22
|
syl5ibrcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = 0 ) ) |
24 |
19 23
|
jaod |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 = 0 ∨ 𝐵 = 0 ) → ( 𝐴 · 𝐵 ) = 0 ) ) |
25 |
15 24
|
impbid |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) ) |