Theorem xdivcld

Description: Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017)

	Ref	Expression
Hypotheses	xdivcld.1	|- ( ph -> A e. RR* )
	xdivcld.2	|- ( ph -> B e. RR )
	xdivcld.3	|- ( ph -> B =/= 0 )
Assertion	xdivcld	|- ( ph -> ( A /e B ) e. RR* )

Proof

Step	Hyp	Ref	Expression
1		xdivcld.1	|- ( ph -> A e. RR* )
2		xdivcld.2	|- ( ph -> B e. RR )
3		xdivcld.3	|- ( ph -> B =/= 0 )
4		xdivval	|- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A /e B ) = ( iota_ x e. RR* ( B *e x ) = A ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A /e B ) = ( iota_ x e. RR* ( B *e x ) = A ) )
6		xreceu	|- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> E! x e. RR* ( B *e x ) = A )
7	1 2 3 6	syl3anc	|- ( ph -> E! x e. RR* ( B *e x ) = A )
8		riotacl	|- ( E! x e. RR* ( B *e x ) = A -> ( iota_ x e. RR* ( B *e x ) = A ) e. RR* )
9	7 8	syl	|- ( ph -> ( iota_ x e. RR* ( B *e x ) = A ) e. RR* )
10	5 9	eqeltrd	|- ( ph -> ( A /e B ) e. RR* )