Theorem subid1

Description: Identity law for subtraction. (Contributed by NM, 9-May-2004) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	subid1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − 0 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		addid1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + 0 ) = 𝐴 )
2	1	oveq1d	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 0 ) − 0 ) = ( 𝐴 − 0 ) )
3		0cn	⊢ 0 ∈ ℂ
4		pncan	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ) → ( ( 𝐴 + 0 ) − 0 ) = 𝐴 )
5	3 4	mpan2	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 0 ) − 0 ) = 𝐴 )
6	2 5	eqtr3d	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − 0 ) = 𝐴 )