Metamath Proof Explorer
Description: A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019)
|
|
Ref |
Expression |
|
Hypotheses |
bj-subcom.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
bj-subcom.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
Assertion |
bj-subcom |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) − ( 𝐵 · 𝐴 ) ) = 0 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
bj-subcom.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
bj-subcom.b |
⊢ ( 𝜑 → 𝐵 ∈ 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 ) |