Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xdivrec Structured version   Visualization version   Unicode version
Theorem xdivrec 29635
Description: Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
Assertion
Ref Expression
xdivrec |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A /𝑒 B ) = ( A xe ( 1 /𝑒 B ) ) )
Proof of Theorem xdivrec
StepHypRef Expression
1 simp2 1062 . . . . . 6 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> B e. RR )
21rexrd 10089 . . . . 5 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> B e. RR* )
3 simp1 1061 . . . . . 6 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> A e. RR* )
4 1re 10039 . . . . . . . . 9 |- 1 e. RR
54rexri 10097 . . . . . . . 8 |- 1 e. RR*
65a1i 11 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> 1 e. RR* )
7 simp3 1063 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> B =/= 0 )
86, 1, 7xdivcld 29631 . . . . . 6 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( 1 /𝑒 B ) e. RR* )
93, 8xmulcld 12132 . . . . 5 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A xe ( 1 /𝑒 B ) ) e. RR* )
10 xmulcom 12096 . . . . 5 |- ( ( B e. RR* /\ ( A xe ( 1 /𝑒 B ) ) e. RR* ) -> ( B xe ( A xe ( 1 /𝑒 B ) ) ) = ( ( A xe ( 1 /𝑒 B ) ) xe B ) )
112, 9, 10syl2anc 693 . . . 4 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( B xe ( A xe ( 1 /𝑒 B ) ) ) = ( ( A xe ( 1 /𝑒 B ) ) xe B ) )
12 xmulass 12117 . . . . 5 |- ( ( A e. RR* /\ ( 1 /𝑒 B ) e. RR* /\ B e. RR* ) -> ( ( A xe ( 1 /𝑒 B ) ) xe B ) = ( A xe ( ( 1 /𝑒 B ) xe B ) ) )
133, 8, 2, 12syl3anc 1326 . . . 4 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( A xe ( 1 /𝑒 B ) ) xe B ) = ( A xe ( ( 1 /𝑒 B ) xe B ) ) )
14 xmulcom 12096 . . . . . . 7 |- ( ( ( 1 /𝑒 B ) e. RR* /\ B e. RR* ) -> ( ( 1 /𝑒 B ) xe B ) = ( B xe ( 1 /𝑒 B ) ) )
158, 2, 14syl2anc 693 . . . . . 6 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( 1 /𝑒 B ) xe B ) = ( B xe ( 1 /𝑒 B ) ) )
16 eqid 2622 . . . . . . 7 |- ( 1 /𝑒 B ) = ( 1 /𝑒 B )
17 xdivmul 29633 . . . . . . . 8 |- ( ( 1 e. RR* /\ ( 1 /𝑒 B ) e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> ( ( 1 /𝑒 B ) = ( 1 /𝑒 B ) <-> ( B xe ( 1 /𝑒 B ) ) = 1 ) )
186, 8, 1, 7, 17syl112anc 1330 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( 1 /𝑒 B ) = ( 1 /𝑒 B ) <-> ( B xe ( 1 /𝑒 B ) ) = 1 ) )
1916, 18mpbii 223 . . . . . 6 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( B xe ( 1 /𝑒 B ) ) = 1 )
2015, 19eqtrd 2656 . . . . 5 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( 1 /𝑒 B ) xe B ) = 1 )
2120oveq2d 6666 . . . 4 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A xe ( ( 1 /𝑒 B ) xe B ) ) = ( A xe 1 ) )
2211, 13, 213eqtrd 2660 . . 3 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( B xe ( A xe ( 1 /𝑒 B ) ) ) = ( A xe 1 ) )
23 xmulid1 12109 . . . 4 |- ( A e. RR* -> ( A xe 1 ) = A )
243, 23syl 17 . . 3 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A xe 1 ) = A )
2522, 24eqtrd 2656 . 2 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( B xe ( A xe ( 1 /𝑒 B ) ) ) = A )
26 xdivmul 29633 . . 3 |- ( ( A e. RR* /\ ( A xe ( 1 /𝑒 B ) ) e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> ( ( A /𝑒 B ) = ( A xe ( 1 /𝑒 B ) ) <-> ( B xe ( A xe ( 1 /𝑒 B ) ) ) = A ) )
273, 9, 1, 7, 26syl112anc 1330 . 2 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( ( A /𝑒 B ) = ( A xe ( 1 /𝑒 B ) ) <-> ( B xe ( A xe ( 1 /𝑒 B ) ) ) = A ) )
2825, 27mpbird 247 1 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A /𝑒 B ) = ( A xe ( 1 /𝑒 B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   xecxmu 11945   /𝑒 cxdiv 29625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-xneg 11946  df-xmul 11948  df-xdiv 29626
This theorem is referenced by:  esumdivc  30145
  Copyright terms: Public domain W3C validator