Step |
Hyp |
Ref |
Expression |
1 |
|
recbothd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
recbothd.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
3 |
|
recbothd.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
4 |
|
recbothd.4 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
5 |
|
recbothd.5 |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
6 |
|
recbothd.6 |
⊢ ( 𝜑 → 𝐶 ≠ 0 ) |
7 |
|
recbothd.7 |
⊢ ( 𝜑 → 𝐷 ∈ ℂ ) |
8 |
|
recbothd.8 |
⊢ ( 𝜑 → 𝐷 ≠ 0 ) |
9 |
1 3 4
|
divcld |
⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ∈ ℂ ) |
10 |
1 3 2 4
|
divne0d |
⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≠ 0 ) |
11 |
9 10
|
jca |
⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ∈ ℂ ∧ ( 𝐴 / 𝐵 ) ≠ 0 ) ) |
12 |
5 7 8
|
divcld |
⊢ ( 𝜑 → ( 𝐶 / 𝐷 ) ∈ ℂ ) |
13 |
5 7 6 8
|
divne0d |
⊢ ( 𝜑 → ( 𝐶 / 𝐷 ) ≠ 0 ) |
14 |
12 13
|
jca |
⊢ ( 𝜑 → ( ( 𝐶 / 𝐷 ) ∈ ℂ ∧ ( 𝐶 / 𝐷 ) ≠ 0 ) ) |
15 |
11 14
|
jca |
⊢ ( 𝜑 → ( ( ( 𝐴 / 𝐵 ) ∈ ℂ ∧ ( 𝐴 / 𝐵 ) ≠ 0 ) ∧ ( ( 𝐶 / 𝐷 ) ∈ ℂ ∧ ( 𝐶 / 𝐷 ) ≠ 0 ) ) ) |
16 |
|
rec11 |
⊢ ( ( ( ( 𝐴 / 𝐵 ) ∈ ℂ ∧ ( 𝐴 / 𝐵 ) ≠ 0 ) ∧ ( ( 𝐶 / 𝐷 ) ∈ ℂ ∧ ( 𝐶 / 𝐷 ) ≠ 0 ) ) → ( ( 1 / ( 𝐴 / 𝐵 ) ) = ( 1 / ( 𝐶 / 𝐷 ) ) ↔ ( 𝐴 / 𝐵 ) = ( 𝐶 / 𝐷 ) ) ) |
17 |
15 16
|
syl |
⊢ ( 𝜑 → ( ( 1 / ( 𝐴 / 𝐵 ) ) = ( 1 / ( 𝐶 / 𝐷 ) ) ↔ ( 𝐴 / 𝐵 ) = ( 𝐶 / 𝐷 ) ) ) |
18 |
17
|
bicomd |
⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) = ( 𝐶 / 𝐷 ) ↔ ( 1 / ( 𝐴 / 𝐵 ) ) = ( 1 / ( 𝐶 / 𝐷 ) ) ) ) |
19 |
1 3 2 4
|
recdivd |
⊢ ( 𝜑 → ( 1 / ( 𝐴 / 𝐵 ) ) = ( 𝐵 / 𝐴 ) ) |
20 |
5 7 6 8
|
recdivd |
⊢ ( 𝜑 → ( 1 / ( 𝐶 / 𝐷 ) ) = ( 𝐷 / 𝐶 ) ) |
21 |
19 20
|
eqeq12d |
⊢ ( 𝜑 → ( ( 1 / ( 𝐴 / 𝐵 ) ) = ( 1 / ( 𝐶 / 𝐷 ) ) ↔ ( 𝐵 / 𝐴 ) = ( 𝐷 / 𝐶 ) ) ) |
22 |
18 21
|
bitrd |
⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) = ( 𝐶 / 𝐷 ) ↔ ( 𝐵 / 𝐴 ) = ( 𝐷 / 𝐶 ) ) ) |