Theorem redivcl

Description: Closure law for division of reals. (Contributed by NM, 27-Sep-1999) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	redivcl	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		simp1	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> A e. RR )
2	1	recnd	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> A e. CC )
3		simp2	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> B e. RR )
4	3	recnd	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> B e. CC )
5		simp3	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> B =/= 0 )
6		divrec	|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
7	2 4 5 6	syl3anc	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
8		rereccl	|- ( ( B e. RR /\ B =/= 0 ) -> ( 1 / B ) e. RR )
9	8	3adant1	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( 1 / B ) e. RR )
10	1 9	remulcld	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A x. ( 1 / B ) ) e. RR )
11	7 10	eqeltrd	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) e. RR )