Theorem subid

Description: Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		addid1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + 0 ) = 𝐴 )
2	1	oveq1d	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 0 ) − 𝐴 ) = ( 𝐴 − 𝐴 ) )
3		0cn	⊢ 0 ∈ ℂ
4		pncan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ) → ( ( 𝐴 + 0 ) − 𝐴 ) = 0 )
5	3 4	mpan2	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 0 ) − 𝐴 ) = 0 )
6	2 5	eqtr3d	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − 𝐴 ) = 0 )