Metamath Proof Explorer


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 ) = ( 𝐵𝐴 ) )

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 ) → - ( 𝐴𝐵 ) = ( - 𝐴 + 𝐵 ) )
5 negsubdi2 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴𝐵 ) = ( 𝐵𝐴 ) )
6 3 4 5 3eqtr2d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 − - 𝐵 ) = ( 𝐵𝐴 ) )