Description: Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | pnpncand.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
pnpncand.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | ||
pnpncand.3 | ⊢ ( 𝜑 → 𝐶 ∈ ℂ ) | ||
Assertion | pnpncand | ⊢ ( 𝜑 → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + ( 𝐶 − 𝐵 ) ) = 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnpncand.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
2 | pnpncand.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | |
3 | pnpncand.3 | ⊢ ( 𝜑 → 𝐶 ∈ ℂ ) | |
4 | 2 3 | subcld | ⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) ∈ ℂ ) |
5 | 1 4 | addcld | ⊢ ( 𝜑 → ( 𝐴 + ( 𝐵 − 𝐶 ) ) ∈ ℂ ) |
6 | 5 2 3 | subsub2d | ⊢ ( 𝜑 → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + ( 𝐶 − 𝐵 ) ) ) |
7 | 1 4 | pncand | ⊢ ( 𝜑 → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) − ( 𝐵 − 𝐶 ) ) = 𝐴 ) |
8 | 6 7 | eqtr3d | ⊢ ( 𝜑 → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + ( 𝐶 − 𝐵 ) ) = 𝐴 ) |