Metamath Proof Explorer


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 )