# Metamath Proof Explorer

## Theorem div2neg

Description: Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012)

Ref Expression
Assertion div2neg ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( - 𝐴 / - 𝐵 ) = ( 𝐴 / 𝐵 ) )

### Proof

Step Hyp Ref Expression
1 negcl ( 𝐵 ∈ ℂ → - 𝐵 ∈ ℂ )
2 1 3ad2ant2 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - 𝐵 ∈ ℂ )
3 simp1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℂ )
4 simp2 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ )
5 simp3 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ≠ 0 )
6 div12 ( ( - 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( - 𝐵 · ( 𝐴 / 𝐵 ) ) = ( 𝐴 · ( - 𝐵 / 𝐵 ) ) )
7 2 3 4 5 6 syl112anc ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( - 𝐵 · ( 𝐴 / 𝐵 ) ) = ( 𝐴 · ( - 𝐵 / 𝐵 ) ) )
8 divneg ( ( 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - ( 𝐵 / 𝐵 ) = ( - 𝐵 / 𝐵 ) )
9 4 8 syld3an1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - ( 𝐵 / 𝐵 ) = ( - 𝐵 / 𝐵 ) )
10 divid ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 / 𝐵 ) = 1 )
11 10 3adant1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 / 𝐵 ) = 1 )
12 11 negeqd ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - ( 𝐵 / 𝐵 ) = - 1 )
13 9 12 eqtr3d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( - 𝐵 / 𝐵 ) = - 1 )
14 13 oveq2d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · ( - 𝐵 / 𝐵 ) ) = ( 𝐴 · - 1 ) )
15 ax-1cn 1 ∈ ℂ
16 15 negcli - 1 ∈ ℂ
17 mulcom ( ( 𝐴 ∈ ℂ ∧ - 1 ∈ ℂ ) → ( 𝐴 · - 1 ) = ( - 1 · 𝐴 ) )
18 16 17 mpan2 ( 𝐴 ∈ ℂ → ( 𝐴 · - 1 ) = ( - 1 · 𝐴 ) )
19 mulm1 ( 𝐴 ∈ ℂ → ( - 1 · 𝐴 ) = - 𝐴 )
20 18 19 eqtrd ( 𝐴 ∈ ℂ → ( 𝐴 · - 1 ) = - 𝐴 )
21 20 3ad2ant1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · - 1 ) = - 𝐴 )
22 14 21 eqtrd ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · ( - 𝐵 / 𝐵 ) ) = - 𝐴 )
23 7 22 eqtrd ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( - 𝐵 · ( 𝐴 / 𝐵 ) ) = - 𝐴 )
24 negcl ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
25 24 3ad2ant1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - 𝐴 ∈ ℂ )
26 divcl ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) ∈ ℂ )
27 negeq0 ( 𝐵 ∈ ℂ → ( 𝐵 = 0 ↔ - 𝐵 = 0 ) )
28 27 necon3bid ( 𝐵 ∈ ℂ → ( 𝐵 ≠ 0 ↔ - 𝐵 ≠ 0 ) )
29 28 biimpa ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - 𝐵 ≠ 0 )
30 29 3adant1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - 𝐵 ≠ 0 )
31 divmul ( ( - 𝐴 ∈ ℂ ∧ ( 𝐴 / 𝐵 ) ∈ ℂ ∧ ( - 𝐵 ∈ ℂ ∧ - 𝐵 ≠ 0 ) ) → ( ( - 𝐴 / - 𝐵 ) = ( 𝐴 / 𝐵 ) ↔ ( - 𝐵 · ( 𝐴 / 𝐵 ) ) = - 𝐴 ) )
32 25 26 2 30 31 syl112anc ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( - 𝐴 / - 𝐵 ) = ( 𝐴 / 𝐵 ) ↔ ( - 𝐵 · ( 𝐴 / 𝐵 ) ) = - 𝐴 ) )
33 23 32 mpbird ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( - 𝐴 / - 𝐵 ) = ( 𝐴 / 𝐵 ) )