Metamath Proof Explorer
Description: Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
negidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
pncand.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
subaddd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
|
Assertion |
pnpcan2d |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐶 ) ) = ( 𝐴 − 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
negidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
pncand.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
subaddd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
4 |
|
pnpcan2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐶 ) ) = ( 𝐴 − 𝐵 ) ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐶 ) ) = ( 𝐴 − 𝐵 ) ) |