Metamath Proof Explorer


Theorem bj-subcom

Description: A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019)

Ref Expression
Hypotheses bj-subcom.a ( 𝜑𝐴 ∈ ℂ )
bj-subcom.b ( 𝜑𝐵 ∈ ℂ )
Assertion bj-subcom ( 𝜑 → ( ( 𝐴 · 𝐵 ) − ( 𝐵 · 𝐴 ) ) = 0 )

Proof

Step Hyp Ref Expression
1 bj-subcom.a ( 𝜑𝐴 ∈ ℂ )
2 bj-subcom.b ( 𝜑𝐵 ∈ ℂ )
3 1 2 mulcld ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℂ )
4 1 2 mulcomd ( 𝜑 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
5 3 4 subeq0bd ( 𝜑 → ( ( 𝐴 · 𝐵 ) − ( 𝐵 · 𝐴 ) ) = 0 )