Theorem divsclw

Description: Weak division closure law. (Contributed by Scott Fenton, 12-Mar-2025)

	Ref	Expression
Assertion	divsclw	Could not format assertion : No typesetting found for |- ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( A /su B ) e. No ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		divsval	Could not format ( ( A e. No /\ B e. No /\ B =/= 0s ) -> ( A /su B ) = ( iota_ y e. No ( B x.s y ) = A ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ B =/= 0s ) -> ( A /su B ) = ( iota_ y e. No ( B x.s y ) = A ) ) with typecode |-
2	1	adantr	Could not format ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( A /su B ) = ( iota_ y e. No ( B x.s y ) = A ) ) : No typesetting found for |- ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( A /su B ) = ( iota_ y e. No ( B x.s y ) = A ) ) with typecode |-
3		3anrot	Could not format ( ( A e. No /\ B e. No /\ B =/= 0s ) <-> ( B e. No /\ B =/= 0s /\ A e. No ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ B =/= 0s ) <-> ( B e. No /\ B =/= 0s /\ A e. No ) ) with typecode |-
4		noreceuw	Could not format ( ( ( B e. No /\ B =/= 0s /\ A e. No ) /\ E. x e. No ( B x.s x ) = 1s ) -> E! y e. No ( B x.s y ) = A ) : No typesetting found for |- ( ( ( B e. No /\ B =/= 0s /\ A e. No ) /\ E. x e. No ( B x.s x ) = 1s ) -> E! y e. No ( B x.s y ) = A ) with typecode |-
5	3 4	sylanb	Could not format ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> E! y e. No ( B x.s y ) = A ) : No typesetting found for |- ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> E! y e. No ( B x.s y ) = A ) with typecode |-
6		riotacl	Could not format ( E! y e. No ( B x.s y ) = A -> ( iota_ y e. No ( B x.s y ) = A ) e. No ) : No typesetting found for |- ( E! y e. No ( B x.s y ) = A -> ( iota_ y e. No ( B x.s y ) = A ) e. No ) with typecode |-
7	5 6	syl	Could not format ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( iota_ y e. No ( B x.s y ) = A ) e. No ) : No typesetting found for |- ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( iota_ y e. No ( B x.s y ) = A ) e. No ) with typecode |-
8	2 7	eqeltrd	Could not format ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( A /su B ) e. No ) : No typesetting found for |- ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( A /su B ) e. No ) with typecode |-