Step |
Hyp |
Ref |
Expression |
1 |
|
mulcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) ) |
2 |
1
|
3adant2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) ) |
3 |
|
mulcom |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
4 |
3
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
5 |
2 4
|
eqeq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ) ) |
6 |
|
mulcan1g |
⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ ( 𝐶 = 0 ∨ 𝐴 = 𝐵 ) ) ) |
7 |
6
|
3coml |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ ( 𝐶 = 0 ∨ 𝐴 = 𝐵 ) ) ) |
8 |
|
orcom |
⊢ ( ( 𝐶 = 0 ∨ 𝐴 = 𝐵 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐶 = 0 ) ) |
9 |
7 8
|
bitrdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐶 = 0 ) ) ) |
10 |
5 9
|
bitrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐶 = 0 ) ) ) |