Theorem abvcl

Description: An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014)

	Ref	Expression
Hypotheses	abvf.a	$$⊢A=\mathrm{AbsVal}\left(R\right)$$
	abvf.b	$$⊢B={\mathrm{Base}}_R$$
Assertion	abvcl	$$⊢\left(F\in A\wedge X\in B\right)\to F\left(X\right)\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		abvf.a	$$⊢A=\mathrm{AbsVal}\left(R\right)$$
2		abvf.b	$$⊢B={\mathrm{Base}}_R$$
3	1 2	abvf	$$⊢F\in A\to F:B⟶ℝ$$
4	3	ffvelrnda	$$⊢\left(F\in A\wedge X\in B\right)\to F\left(X\right)\in ℝ$$