Theorem pntrf

Description: Functionality of the residual. Lemma for pnt . (Contributed by Mario Carneiro, 8-Apr-2016)

	Ref	Expression
Hypothesis	pntrval.r	$$⊢R=\left(a\in {ℝ}^{+}⟼\psi \left(a\right)-a\right)$$
Assertion	pntrf	$$⊢R:{ℝ}^{+}⟶ℝ$$

Proof

Step	Hyp	Ref	Expression
1		pntrval.r	$$⊢R=\left(a\in {ℝ}^{+}⟼\psi \left(a\right)-a\right)$$
2		rpre	$$⊢a\in {ℝ}^{+}\to a\in ℝ$$
3		chpcl	$$⊢a\in ℝ\to \psi \left(a\right)\in ℝ$$
4	2 3	syl	$$⊢a\in {ℝ}^{+}\to \psi \left(a\right)\in ℝ$$
5	4 2	resubcld	$$⊢a\in {ℝ}^{+}\to \psi \left(a\right)-a\in ℝ$$
6	1 5	fmpti	$$⊢R:{ℝ}^{+}⟶ℝ$$