Theorem subrfn

Description: Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012)

	Ref	Expression
Assertion	subrfn	$$⊢\left(A\in C\wedge B\in D\right)\to \left(A{-}_{r}B\right)Fnℝ$$

Proof

Step	Hyp	Ref	Expression
1		ovex	$$⊢A\left(x\right)-B\left(x\right)\in \mathrm{V}$$
2		eqid	$$⊢\left(x\in ℝ⟼A\left(x\right)-B\left(x\right)\right)=\left(x\in ℝ⟼A\left(x\right)-B\left(x\right)\right)$$
3	1 2	fnmpti	$$⊢\left(x\in ℝ⟼A\left(x\right)-B\left(x\right)\right)Fnℝ$$
4		subrval	$$⊢\left(A\in C\wedge B\in D\right)\to A{-}_{r}B=\left(x\in ℝ⟼A\left(x\right)-B\left(x\right)\right)$$
5	4	fneq1d	$$⊢\left(A\in C\wedge B\in D\right)\to \left(\left(A{-}_{r}B\right)Fnℝ\leftrightarrow \left(x\in ℝ⟼A\left(x\right)-B\left(x\right)\right)Fnℝ\right)$$
6	3 5	mpbiri	$$⊢\left(A\in C\wedge B\in D\right)\to \left(A{-}_{r}B\right)Fnℝ$$