Theorem rersubcl

Description: Closure for real subtraction. Based on subcl . (Contributed by Steven Nguyen, 7-Jan-2023)

	Ref	Expression
Assertion	rersubcl	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A{-}_{ℝ}B\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		resubval	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A{-}_{ℝ}B=\left(\iota x\in ℝ|B+x=A\right)$$
2		resubeu	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \exists !x\in ℝB+x=A$$
3	2	ancoms	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \exists !x\in ℝB+x=A$$
4		riotacl	$$⊢\exists !x\in ℝB+x=A\to \left(\iota x\in ℝ|B+x=A\right)\in ℝ$$
5	3 4	syl	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(\iota x\in ℝ|B+x=A\right)\in ℝ$$
6	1 5	eqeltrd	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A{-}_{ℝ}B\in ℝ$$