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 ) )