Theorem neg2sub

Description: Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007)

	Ref	Expression
Assertion	neg2sub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 − - 𝐵 ) = ( 𝐵 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
2		subneg	⊢ ( ( - 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 − - 𝐵 ) = ( - 𝐴 + 𝐵 ) )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 − - 𝐵 ) = ( - 𝐴 + 𝐵 ) )
4		negsubdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( - 𝐴 + 𝐵 ) )
5		negsubdi2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐴 ) )
6	3 4 5	3eqtr2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 − - 𝐵 ) = ( 𝐵 − 𝐴 ) )