Theorem xrltled

Description: 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle . (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	xrltled.a	$$⊢\phi \to A\in {ℝ}^{*}$$
	xrltled.b	$$⊢\phi \to B\in {ℝ}^{*}$$
	xrltled.altb	$$⊢\phi \to A<B$$
Assertion	xrltled	$$⊢\phi \to A\le B$$

Proof

Step	Hyp	Ref	Expression
1		xrltled.a	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xrltled.b	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xrltled.altb	$$⊢\phi \to A<B$$
4		xrltle	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A<B\to A\le B\right)$$
5	1 2 4	syl2anc	$$⊢\phi \to \left(A<B\to A\le B\right)$$
6	3 5	mpd	$$⊢\phi \to A\le B$$