Metamath Proof Explorer
Description: Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 11-Oct-2018)
|
|
Ref |
Expression |
|
Hypotheses |
joinlmulsubmuli.1 |
⊢ 𝐴 ∈ ℂ |
|
|
joinlmulsubmuli.2 |
⊢ 𝐵 ∈ ℂ |
|
|
joinlmulsubmuli.3 |
⊢ 𝐶 ∈ ℂ |
|
|
joinlmulsubmuli.4 |
⊢ ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) ) = 𝐷 |
|
Assertion |
joinlmulsubmuli |
⊢ ( ( 𝐴 − 𝐶 ) · 𝐵 ) = 𝐷 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
joinlmulsubmuli.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
joinlmulsubmuli.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
joinlmulsubmuli.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
joinlmulsubmuli.4 |
⊢ ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) ) = 𝐷 |
5 |
1 3 2
|
subdiri |
⊢ ( ( 𝐴 − 𝐶 ) · 𝐵 ) = ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) ) |
6 |
5 4
|
eqtri |
⊢ ( ( 𝐴 − 𝐶 ) · 𝐵 ) = 𝐷 |