Theorem npcan

Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	npcan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		subcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − 𝐵 ) ∈ ℂ )
2		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ )
3	1 2	addcomd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + 𝐵 ) = ( 𝐵 + ( 𝐴 − 𝐵 ) ) )
4		pncan3	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 + ( 𝐴 − 𝐵 ) ) = 𝐴 )
5	4	ancoms	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 + ( 𝐴 − 𝐵 ) ) = 𝐴 )
6	3 5	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + 𝐵 ) = 𝐴 )