Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐴 ∈ ℂ ) |
2 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐵 ∈ ℂ ) |
3 |
|
reccl |
⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 1 / 𝐶 ) ∈ ℂ ) |
4 |
3
|
3ad2ant3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 1 / 𝐶 ) ∈ ℂ ) |
5 |
1 2 4
|
adddird |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 + 𝐵 ) · ( 1 / 𝐶 ) ) = ( ( 𝐴 · ( 1 / 𝐶 ) ) + ( 𝐵 · ( 1 / 𝐶 ) ) ) ) |
6 |
1 2
|
addcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 + 𝐵 ) ∈ ℂ ) |
7 |
|
simp3l |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐶 ∈ ℂ ) |
8 |
|
simp3r |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐶 ≠ 0 ) |
9 |
|
divrec |
⊢ ( ( ( 𝐴 + 𝐵 ) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( ( 𝐴 + 𝐵 ) / 𝐶 ) = ( ( 𝐴 + 𝐵 ) · ( 1 / 𝐶 ) ) ) |
10 |
6 7 8 9
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 + 𝐵 ) / 𝐶 ) = ( ( 𝐴 + 𝐵 ) · ( 1 / 𝐶 ) ) ) |
11 |
|
divrec |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐴 / 𝐶 ) = ( 𝐴 · ( 1 / 𝐶 ) ) ) |
12 |
1 7 8 11
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / 𝐶 ) = ( 𝐴 · ( 1 / 𝐶 ) ) ) |
13 |
|
divrec |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐵 / 𝐶 ) = ( 𝐵 · ( 1 / 𝐶 ) ) ) |
14 |
2 7 8 13
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐵 / 𝐶 ) = ( 𝐵 · ( 1 / 𝐶 ) ) ) |
15 |
12 14
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) + ( 𝐵 / 𝐶 ) ) = ( ( 𝐴 · ( 1 / 𝐶 ) ) + ( 𝐵 · ( 1 / 𝐶 ) ) ) ) |
16 |
5 10 15
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 + 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) + ( 𝐵 / 𝐶 ) ) ) |