Theorem negsubdi

Description: Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	negsubdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( - 𝐴 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		0cn	⊢ 0 ∈ ℂ
2		subsub	⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 0 − ( 𝐴 − 𝐵 ) ) = ( ( 0 − 𝐴 ) + 𝐵 ) )
3	1 2	mp3an1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 0 − ( 𝐴 − 𝐵 ) ) = ( ( 0 − 𝐴 ) + 𝐵 ) )
4		df-neg	⊢ - ( 𝐴 − 𝐵 ) = ( 0 − ( 𝐴 − 𝐵 ) )
5		df-neg	⊢ - 𝐴 = ( 0 − 𝐴 )
6	5	oveq1i	⊢ ( - 𝐴 + 𝐵 ) = ( ( 0 − 𝐴 ) + 𝐵 )
7	3 4 6	3eqtr4g	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( - 𝐴 + 𝐵 ) )