Metamath Proof Explorer
Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006)
|
|
Ref |
Expression |
|
Hypotheses |
negidi.1 |
⊢ 𝐴 ∈ ℂ |
|
|
pncan3i.2 |
⊢ 𝐵 ∈ ℂ |
|
|
subadd.3 |
⊢ 𝐶 ∈ ℂ |
|
Assertion |
pnncani |
⊢ ( ( 𝐴 + 𝐵 ) − ( 𝐴 − 𝐶 ) ) = ( 𝐵 + 𝐶 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
negidi.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
pncan3i.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
subadd.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
pnncan |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − ( 𝐴 − 𝐶 ) ) = ( 𝐵 + 𝐶 ) ) |
5 |
1 2 3 4
|
mp3an |
⊢ ( ( 𝐴 + 𝐵 ) − ( 𝐴 − 𝐶 ) ) = ( 𝐵 + 𝐶 ) |