Theorem divsmulwd

Description: Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. (Contributed by Scott Fenton, 12-Mar-2025)

	Ref	Expression
Hypotheses	divsmulwd.1	$$⊢\phi \to A\in \mathrm{No}$$
	divsmulwd.2	$$⊢\phi \to B\in \mathrm{No}$$
	divsmulwd.3	$$⊢\phi \to C\in \mathrm{No}$$
	divsmulwd.4	No typesetting found for |- ( ph -> C =/= 0s ) with typecode |-
	divsmulwd.5	No typesetting found for |- ( ph -> E. x e. No ( C x.s x ) = 1s ) with typecode |-
Assertion	divsmulwd	Could not format assertion : No typesetting found for |- ( ph -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		divsmulwd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divsmulwd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divsmulwd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		divsmulwd.4	Could not format ( ph -> C =/= 0s ) : No typesetting found for |- ( ph -> C =/= 0s ) with typecode |-
5		divsmulwd.5	Could not format ( ph -> E. x e. No ( C x.s x ) = 1s ) : No typesetting found for |- ( ph -> E. x e. No ( C x.s x ) = 1s ) with typecode |-
6	3 4	jca	Could not format ( ph -> ( C e. No /\ C =/= 0s ) ) : No typesetting found for |- ( ph -> ( C e. No /\ C =/= 0s ) ) with typecode |-
7		divsmulw	Could not format ( ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) /\ E. x e. No ( C x.s x ) = 1s ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) : No typesetting found for |- ( ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) /\ E. x e. No ( C x.s x ) = 1s ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) with typecode |-
8	1 2 6 5 7	syl31anc	Could not format ( ph -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) : No typesetting found for |- ( ph -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) with typecode |-