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	Could not format assertion : No typesetting found for |- ( ( A e. No /\ A =/= 0s ) -> E* x e. No ( A x.s x ) = B ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		eqtr3	Could not format ( ( ( A x.s x ) = B /\ ( A x.s y ) = B ) -> ( A x.s x ) = ( A x.s y ) ) : No typesetting found for |- ( ( ( A x.s x ) = B /\ ( A x.s y ) = B ) -> ( A x.s x ) = ( A x.s y ) ) with typecode |-
2		simprl	Could not format ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> x e. No ) : No typesetting found for |- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> x e. No ) with typecode |-
3		simprr	Could not format ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> y e. No ) : No typesetting found for |- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> y e. No ) with typecode |-
4		simpll	Could not format ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> A e. No ) : No typesetting found for |- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> A e. No ) with typecode |-
5		simplr	Could not format ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> A =/= 0s ) : No typesetting found for |- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> A =/= 0s ) with typecode |-
6	2 3 4 5	mulscan1d	Could not format ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> ( ( A x.s x ) = ( A x.s y ) <-> x = y ) ) : No typesetting found for |- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> ( ( A x.s x ) = ( A x.s y ) <-> x = y ) ) with typecode |-
7	1 6	imbitrid	Could not format ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> ( ( ( A x.s x ) = B /\ ( A x.s y ) = B ) -> x = y ) ) : No typesetting found for |- ( ( ( A e. No /\ A =/= 0s ) /\ ( x e. No /\ y e. No ) ) -> ( ( ( A x.s x ) = B /\ ( A x.s y ) = B ) -> x = y ) ) with typecode |-
8	7	ralrimivva	Could not format ( ( 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 ) ) : No typesetting found for |- ( ( 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 ) ) with typecode |-
9		oveq2	Could not format ( x = y -> ( A x.s x ) = ( A x.s y ) ) : No typesetting found for |- ( x = y -> ( A x.s x ) = ( A x.s y ) ) with typecode |-
10	9	eqeq1d	Could not format ( x = y -> ( ( A x.s x ) = B <-> ( A x.s y ) = B ) ) : No typesetting found for |- ( x = y -> ( ( A x.s x ) = B <-> ( A x.s y ) = B ) ) with typecode |-
11	10	rmo4	Could not format ( 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 ) ) : No typesetting found for |- ( 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 ) ) with typecode |-
12	8 11	sylibr	Could not format ( ( A e. No /\ A =/= 0s ) -> E* x e. No ( A x.s x ) = B ) : No typesetting found for |- ( ( A e. No /\ A =/= 0s ) -> E* x e. No ( A x.s x ) = B ) with typecode |-