Step |
Hyp |
Ref |
Expression |
1 |
|
mulneg2 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · - 𝐶 ) = - ( 𝐵 · 𝐶 ) ) |
2 |
1
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ) → ( 𝐵 · - 𝐶 ) = - ( 𝐵 · 𝐶 ) ) |
3 |
2
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ) → ( 𝐴 + ( 𝐵 · - 𝐶 ) ) = ( 𝐴 + - ( 𝐵 · 𝐶 ) ) ) |
4 |
|
mulcl |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) ∈ ℂ ) |
5 |
|
negsub |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 · 𝐶 ) ∈ ℂ ) → ( 𝐴 + - ( 𝐵 · 𝐶 ) ) = ( 𝐴 − ( 𝐵 · 𝐶 ) ) ) |
6 |
4 5
|
sylan2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ) → ( 𝐴 + - ( 𝐵 · 𝐶 ) ) = ( 𝐴 − ( 𝐵 · 𝐶 ) ) ) |
7 |
3 6
|
eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ) → ( 𝐴 − ( 𝐵 · 𝐶 ) ) = ( 𝐴 + ( 𝐵 · - 𝐶 ) ) ) |
8 |
7
|
3impb |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 · 𝐶 ) ) = ( 𝐴 + ( 𝐵 · - 𝐶 ) ) ) |