Theorem xleadd2d

Description: Addition of extended reals preserves the "less than or equal to" relation, in the right slot. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xleadd2d.1	$$⊢\phi \to A\in {ℝ}^{*}$$
	xleadd2d.2	$$⊢\phi \to B\in {ℝ}^{*}$$
	xleadd2d.3	$$⊢\phi \to C\in {ℝ}^{*}$$
	xleadd2d.4	$$⊢\phi \to A\le B$$
Assertion	xleadd2d	$$⊢\phi \to C{+}_{𝑒}A\le C{+}_{𝑒}B$$

Proof

Step	Hyp	Ref	Expression
1		xleadd2d.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xleadd2d.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xleadd2d.3	$$⊢\phi \to C\in {ℝ}^{*}$$
4		xleadd2d.4	$$⊢\phi \to A\le B$$
5		xleadd2a	$$⊢\left(\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\wedge A\le B\right)\to C{+}_{𝑒}A\le C{+}_{𝑒}B$$
6	1 2 3 4 5	syl31anc	$$⊢\phi \to C{+}_{𝑒}A\le C{+}_{𝑒}B$$