Theorem xleadd1d

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

	Ref	Expression
Hypotheses	xleadd1d.1	$$⊢\phi \to A\in {ℝ}^{*}$$
	xleadd1d.2	$$⊢\phi \to B\in {ℝ}^{*}$$
	xleadd1d.3	$$⊢\phi \to C\in {ℝ}^{*}$$
	xleadd1d.4	$$⊢\phi \to A\le B$$
Assertion	xleadd1d	$$⊢\phi \to A{+}_{𝑒}C\le B{+}_{𝑒}C$$

Proof

Step	Hyp	Ref	Expression
1		xleadd1d.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xleadd1d.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xleadd1d.3	$$⊢\phi \to C\in {ℝ}^{*}$$
4		xleadd1d.4	$$⊢\phi \to A\le B$$
5		xleadd1a	$$⊢\left(\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\wedge A\le B\right)\to A{+}_{𝑒}C\le B{+}_{𝑒}C$$
6	1 2 3 4 5	syl31anc	$$⊢\phi \to A{+}_{𝑒}C\le B{+}_{𝑒}C$$