Theorem 0pval

Description: The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014)

	Ref	Expression
Assertion	0pval	$$⊢A\in ℂ\to 0_{𝑝}\left(A\right)=0$$

Proof

Step	Hyp	Ref	Expression
1		df-0p	$$⊢0_{𝑝}=ℂ\times \left\{0\right\}$$
2	1	fveq1i	$$⊢0_{𝑝}\left(A\right)=\left(ℂ\times \left\{0\right\}\right)\left(A\right)$$
3		c0ex	$$⊢0\in \mathrm{V}$$
4	3	fvconst2	$$⊢A\in ℂ\to \left(ℂ\times \left\{0\right\}\right)\left(A\right)=0$$
5	2 4	eqtrid	$$⊢A\in ℂ\to 0_{𝑝}\left(A\right)=0$$