Theorem subneg

Description: Relationship between subtraction and negative. (Contributed by NM, 10-May-2004) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		df-neg	⊢ - 𝐵 = ( 0 − 𝐵 )
2	1	oveq2i	⊢ ( 𝐴 − - 𝐵 ) = ( 𝐴 − ( 0 − 𝐵 ) )
3		0cn	⊢ 0 ∈ ℂ
4		subsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − ( 0 − 𝐵 ) ) = ( ( 𝐴 − 0 ) + 𝐵 ) )
5	3 4	mp3an2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − ( 0 − 𝐵 ) ) = ( ( 𝐴 − 0 ) + 𝐵 ) )
6	2 5	syl5eq	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − - 𝐵 ) = ( ( 𝐴 − 0 ) + 𝐵 ) )
7		subid1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − 0 ) = 𝐴 )
8	7	adantr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − 0 ) = 𝐴 )
9	8	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 0 ) + 𝐵 ) = ( 𝐴 + 𝐵 ) )
10	6 9	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − - 𝐵 ) = ( 𝐴 + 𝐵 ) )