Step |
Hyp |
Ref |
Expression |
1 |
|
divmul2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ 𝐴 = ( 𝐶 · 𝐵 ) ) ) |
2 |
|
mulcom |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
3 |
2
|
adantrr |
⊢ ( ( 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
4 |
3
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
5 |
4
|
eqeq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 = ( 𝐵 · 𝐶 ) ↔ 𝐴 = ( 𝐶 · 𝐵 ) ) ) |
6 |
1 5
|
bitr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ 𝐴 = ( 𝐵 · 𝐶 ) ) ) |