Theorem divsassd

Description: An associative law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025)

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

Proof

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