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	$$⊢A\in ℂ\to A-A=0$$

Proof

Step	Hyp	Ref	Expression
1		addrid	$$⊢A\in ℂ\to A+0=A$$
2	1	oveq1d	$$⊢A\in ℂ\to A+0-A=A-A$$
3		0cn	$$⊢0\in ℂ$$
4		pncan2	$$⊢\left(A\in ℂ\wedge 0\in ℂ\right)\to A+0-A=0$$
5	3 4	mpan2	$$⊢A\in ℂ\to A+0-A=0$$
6	2 5	eqtr3d	$$⊢A\in ℂ\to A-A=0$$