Theorem divsmo

Description: Uniqueness of surreal inversion. Given a non-zero surreal A , there is at most one surreal giving a particular product. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Assertion	divsmo	|- ( ( A e. No /\ A =/= 0s ) -> E* x e. No ( A x.s x ) = B )

Proof

Step	Hyp	Ref	Expression
1		eqtr3	|- ( ( ( A x.s x ) = B /\ ( A x.s y ) = B ) -> ( A x.s x ) = ( A x.s y ) )
2		simprl	|- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> x e. No )
3		simprr	|- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> y e. No )
4		simpll	|- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> A e. No )
5		simplr	|- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> A =/= 0s )
6	2 3 4 5	mulscan1d	|- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> ( ( A x.s x ) = ( A x.s y ) <-> x = y ) )
7	1 6	imbitrid	|- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> ( ( ( A x.s x ) = B /\ ( A x.s y ) = B ) -> x = y ) )
8	7	ralrimivva	|- ( ( A e. No /\ A =/= 0s ) -> A. x e. No A. y e. No ( ( ( A x.s x ) = B /\ ( A x.s y ) = B ) -> x = y ) )
9		oveq2	|- ( x = y -> ( A x.s x ) = ( A x.s y ) )
10	9	eqeq1d	|- ( x = y -> ( ( A x.s x ) = B <-> ( A x.s y ) = B ) )
11	10	rmo4	|- ( E* x e. No ( A x.s x ) = B <-> A. x e. No A. y e. No ( ( ( A x.s x ) = B /\ ( A x.s y ) = B ) -> x = y ) )
12	8 11	sylibr	|- ( ( A e. No /\ A =/= 0s ) -> E* x e. No ( A x.s x ) = B )