Theorem addneg1mul

Description: Addition with product with minus one is a subtraction. (Contributed by AV, 18-Oct-2021)

	Ref	Expression
Assertion	addneg1mul	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + ( - 1 · 𝐵 ) ) = ( 𝐴 − 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		mulm1	⊢ ( 𝐵 ∈ ℂ → ( - 1 · 𝐵 ) = - 𝐵 )
2	1	adantl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 1 · 𝐵 ) = - 𝐵 )
3	2	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + ( - 1 · 𝐵 ) ) = ( 𝐴 + - 𝐵 ) )
4		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
5	3 4	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + ( - 1 · 𝐵 ) ) = ( 𝐴 − 𝐵 ) )