Theorem divsmuld

Description: Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	divsmuld.1	|- ( ph -> A e. No )
	divsmuld.2	|- ( ph -> B e. No )
	divsmuld.3	|- ( ph -> C e. No )
	divsmuld.4	|- ( ph -> C =/= 0s )
Assertion	divsmuld	|- ( ph -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		divsmuld.1	|- ( ph -> A e. No )
2		divsmuld.2	|- ( ph -> B e. No )
3		divsmuld.3	|- ( ph -> C e. No )
4		divsmuld.4	|- ( ph -> C =/= 0s )
5	3 4	recsexd	|- ( ph -> E. x e. No ( C x.s x ) = 1s )
6	1 2 3 4 5	divsmulwd	|- ( ph -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) )