Theorem mulscl

Description: The surreals are closed under multiplication. Theorem 8(i) of Conway p. 19. (Contributed by Scott Fenton, 5-Mar-2025)

	Ref	Expression
Assertion	mulscl	Could not format assertion : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A x.s B ) e. No ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		0sno	Could not format 0s e. No : No typesetting found for |- 0s e. No with typecode |-
2	1 1	pm3.2i	Could not format ( 0s e. No /\ 0s e. No ) : No typesetting found for |- ( 0s e. No /\ 0s e. No ) with typecode |-
3		mulsprop	Could not format ( ( ( A e. No /\ B e. No ) /\ ( 0s e. No /\ 0s e. No ) /\ ( 0s e. No /\ 0s e. No ) ) -> ( ( A x.s B ) e. No /\ ( ( 0s ( ( 0s x.s 0s ) -s ( 0s x.s 0s ) ) ( ( A x.s B ) e. No /\ ( ( 0s ( ( 0s x.s 0s ) -s ( 0s x.s 0s ) )
4	2 2 3	mp3an23	Could not format ( ( A e. No /\ B e. No ) -> ( ( A x.s B ) e. No /\ ( ( 0s ( ( 0s x.s 0s ) -s ( 0s x.s 0s ) ) ( ( A x.s B ) e. No /\ ( ( 0s ( ( 0s x.s 0s ) -s ( 0s x.s 0s ) )
5	4	simpld	Could not format ( ( A e. No /\ B e. No ) -> ( A x.s B ) e. No ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A x.s B ) e. No ) with typecode |-