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	Could not format assertion : No typesetting found for |- ( ph -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) ) with typecode |-

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	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) ) with typecode |-
5	1 2 3 4	syl3anc	Could not format ( ph -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) ) : No typesetting found for |- ( ph -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) ) with typecode |-