Step |
Hyp |
Ref |
Expression |
1 |
|
ax-icn |
⊢ i ∈ ℂ |
2 |
1
|
mul01i |
⊢ ( i · 0 ) = 0 |
3 |
2
|
oveq2i |
⊢ ( 0 + ( i · 0 ) ) = ( 0 + 0 ) |
4 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
5 |
3 4
|
eqtri |
⊢ ( 0 + ( i · 0 ) ) = 0 |
6 |
5
|
eqeq2i |
⊢ ( ( 𝐴 + ( i · 𝐵 ) ) = ( 0 + ( i · 0 ) ) ↔ ( 𝐴 + ( i · 𝐵 ) ) = 0 ) |
7 |
|
0re |
⊢ 0 ∈ ℝ |
8 |
|
cru |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ∈ ℝ ∧ 0 ∈ ℝ ) ) → ( ( 𝐴 + ( i · 𝐵 ) ) = ( 0 + ( i · 0 ) ) ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) |
9 |
7 7 8
|
mpanr12 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + ( i · 𝐵 ) ) = ( 0 + ( i · 0 ) ) ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) |
10 |
6 9
|
bitr3id |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + ( i · 𝐵 ) ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) |
11 |
10
|
necon3abid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + ( i · 𝐵 ) ) ≠ 0 ↔ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) |
12 |
|
neorian |
⊢ ( ( 𝐴 ≠ 0 ∨ 𝐵 ≠ 0 ) ↔ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) |
13 |
11 12
|
syl6rbbr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 ≠ 0 ∨ 𝐵 ≠ 0 ) ↔ ( 𝐴 + ( i · 𝐵 ) ) ≠ 0 ) ) |