Theorem addscld

Description: Surreal numbers are closed under addition. Theorem 6(iii) of Conway p. 18. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Hypotheses	addscut.1	|- ( ph -> X e. No )
	addscut.2	|- ( ph -> Y e. No )
Assertion	addscld	|- ( ph -> ( X +s Y ) e. No )

Proof

Step	Hyp	Ref	Expression
1		addscut.1	|- ( ph -> X e. No )
2		addscut.2	|- ( ph -> Y e. No )
3	1 2	addscut	|- ( ph -> ( ( X +s Y ) e. No /\ ( { p | E. l e. ( _Left ` X ) p = ( l +s Y ) } u. { q | E. m e. ( _Left ` Y ) q = ( X +s m ) } ) <
4	3	simp1d	|- ( ph -> ( X +s Y ) e. No )