Metamath Proof Explorer

Theorem xdivrec

Description: Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017)

Ref Expression
Assertion xdivrec 
|- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A /e B ) = ( A *e ( 1 /e B ) ) )

Proof

Step Hyp Ref Expression
1 simp2 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> B e. RR )
2 1 rexrd 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> B e. RR* )
3 simp1 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> A e. RR* )
4 1xr 
 |-  1 e. RR*
5 4 a1i 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> 1 e. RR* )
6 simp3 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> B =/= 0 )
7 5 1 6 xdivcld 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( 1 /e B ) e. RR* )
8 3 7 xmulcld 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A *e ( 1 /e B ) ) e. RR* )
9 xmulcom 
 |-  ( ( B e. RR* /\ ( A *e ( 1 /e B ) ) e. RR* ) -> ( B *e ( A *e ( 1 /e B ) ) ) = ( ( A *e ( 1 /e B ) ) *e B ) )
10 2 8 9 syl2anc 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( B *e ( A *e ( 1 /e B ) ) ) = ( ( A *e ( 1 /e B ) ) *e B ) )
11 xmulass 
 |-  ( ( A e. RR* /\ ( 1 /e B ) e. RR* /\ B e. RR* ) -> ( ( A *e ( 1 /e B ) ) *e B ) = ( A *e ( ( 1 /e B ) *e B ) ) )
12 3 7 2 11 syl3anc 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( A *e ( 1 /e B ) ) *e B ) = ( A *e ( ( 1 /e B ) *e B ) ) )
13 xmulcom 
 |-  ( ( ( 1 /e B ) e. RR* /\ B e. RR* ) -> ( ( 1 /e B ) *e B ) = ( B *e ( 1 /e B ) ) )
14 7 2 13 syl2anc 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( 1 /e B ) *e B ) = ( B *e ( 1 /e B ) ) )
15 eqid 
 |-  ( 1 /e B ) = ( 1 /e B )
16 xdivmul 
 |-  ( ( 1 e. RR* /\ ( 1 /e B ) e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> ( ( 1 /e B ) = ( 1 /e B ) <-> ( B *e ( 1 /e B ) ) = 1 ) )
17 5 7 1 6 16 syl112anc 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( 1 /e B ) = ( 1 /e B ) <-> ( B *e ( 1 /e B ) ) = 1 ) )
18 15 17 mpbii 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( B *e ( 1 /e B ) ) = 1 )
19 14 18 eqtrd 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( 1 /e B ) *e B ) = 1 )
20 19 oveq2d 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A *e ( ( 1 /e B ) *e B ) ) = ( A *e 1 ) )
21 10 12 20 3eqtrd 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( B *e ( A *e ( 1 /e B ) ) ) = ( A *e 1 ) )
22 xmulid1 
 |-  ( A e. RR* -> ( A *e 1 ) = A )
23 3 22 syl 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A *e 1 ) = A )
24 21 23 eqtrd 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( B *e ( A *e ( 1 /e B ) ) ) = A )
25 xdivmul 
 |-  ( ( A e. RR* /\ ( A *e ( 1 /e B ) ) e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> ( ( A /e B ) = ( A *e ( 1 /e B ) ) <-> ( B *e ( A *e ( 1 /e B ) ) ) = A ) )
26 3 8 1 6 25 syl112anc 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( A /e B ) = ( A *e ( 1 /e B ) ) <-> ( B *e ( A *e ( 1 /e B ) ) ) = A ) )
27 24 26 mpbird 
 |-  ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A /e B ) = ( A *e ( 1 /e B ) ) )