Theorem sleadd1d

Description: Addition to both sides of surreal less-than or equal. Theorem 5 of Conway p. 18. (Contributed by Scott Fenton, 21-Jan-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	sleadd1d	$$⊢\phi \to \left(A{\le }_{s}B\leftrightarrow \left(A{+}_{s}C\right){\le }_s\left(B{+}_{s}C\right)\right)$$

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		sleadd1	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to \left(A{\le }_{s}B\leftrightarrow \left(A{+}_{s}C\right){\le }_s\left(B{+}_{s}C\right)\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A{\le }_{s}B\leftrightarrow \left(A{+}_{s}C\right){\le }_s\left(B{+}_{s}C\right)\right)$$