Theorem sleadd2d

Description: Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	addscand.1	$$⊢\phi \to A\in \mathrm{No}$$
	addscand.2	$$⊢\phi \to B\in \mathrm{No}$$
	addscand.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	sleadd2d	$$⊢\phi \to \left(A{\le }_{s}B\leftrightarrow \left(C{+}_{s}A\right){\le }_s\left(C{+}_{s}B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		addscand.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addscand.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addscand.3	$$⊢\phi \to C\in \mathrm{No}$$
4		sleadd2	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to \left(A{\le }_{s}B\leftrightarrow \left(C{+}_{s}A\right){\le }_s\left(C{+}_{s}B\right)\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A{\le }_{s}B\leftrightarrow \left(C{+}_{s}A\right){\le }_s\left(C{+}_{s}B\right)\right)$$