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Metamath Proof Explorer

Theorem divneg2

Description: Move negative sign inside of a division. (Contributed by Mario Carneiro, 15-Sep-2014)

Ref Expression
Assertion divneg2 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - ( 𝐴 / 𝐵 ) = ( 𝐴 / - 𝐵 ) )

Proof

Step Hyp Ref Expression
1 divneg ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - ( 𝐴 / 𝐵 ) = ( - 𝐴 / 𝐵 ) )
2 negcl ⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
3 div2neg ⊢ ( ( - 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( - - 𝐴 / - 𝐵 ) = ( - 𝐴 / 𝐵 ) )
4 2 3 syl3an1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( - - 𝐴 / - 𝐵 ) = ( - 𝐴 / 𝐵 ) )
5 negneg ⊢ ( 𝐴 ∈ ℂ → - - 𝐴 = 𝐴 )
6 5 3ad2ant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - - 𝐴 = 𝐴 )
7 6 oveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( - - 𝐴 / - 𝐵 ) = ( 𝐴 / - 𝐵 ) )
8 1 4 7 3eqtr2d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → - ( 𝐴 / 𝐵 ) = ( 𝐴 / - 𝐵 ) )