Metamath Proof Explorer
Description: Move the left term in a sum on the LHS to the RHS. (Contributed by David
A. Wheeler, 11-Oct-2018)
|
|
Ref |
Expression |
|
Hypotheses |
mvlladdi.1 |
⊢ 𝐴 ∈ ℂ |
|
|
mvlladdi.2 |
⊢ 𝐵 ∈ ℂ |
|
|
mvlladdi.3 |
⊢ ( 𝐴 + 𝐵 ) = 𝐶 |
|
Assertion |
mvlladdi |
⊢ 𝐵 = ( 𝐶 − 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mvlladdi.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
mvlladdi.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
mvlladdi.3 |
⊢ ( 𝐴 + 𝐵 ) = 𝐶 |
4 |
2 1
|
pncan3oi |
⊢ ( ( 𝐵 + 𝐴 ) − 𝐴 ) = 𝐵 |
5 |
1 2 3
|
addcomli |
⊢ ( 𝐵 + 𝐴 ) = 𝐶 |
6 |
5
|
oveq1i |
⊢ ( ( 𝐵 + 𝐴 ) − 𝐴 ) = ( 𝐶 − 𝐴 ) |
7 |
4 6
|
eqtr3i |
⊢ 𝐵 = ( 𝐶 − 𝐴 ) |