Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xdivmul Structured version   Visualization version   Unicode version

Theorem xdivmul 28387
Description: Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
Assertion
Ref Expression
xdivmul  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( ( A /𝑒  C )  =  B  <->  ( C xe B )  =  A ) )

Proof of Theorem xdivmul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xdivval 28381 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( A /𝑒  C )  =  (
iota_ x  e.  RR*  ( C xe x )  =  A ) )
213expb 1208 . . . 4  |-  ( ( A  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( A /𝑒  C )  =  (
iota_ x  e.  RR*  ( C xe x )  =  A ) )
323adant2 1026 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( A /𝑒  C )  =  (
iota_ x  e.  RR*  ( C xe x )  =  A ) )
43eqeq1d 2452 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( ( A /𝑒  C )  =  B  <->  ( iota_ x  e.  RR*  ( C xe x )  =  A )  =  B ) )
5 simp2 1008 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  B  e.  RR* )
6 xreceu 28384 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  E! x  e.  RR*  ( C xe x )  =  A )
763expb 1208 . . . 4  |-  ( ( A  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  E! x  e.  RR*  ( C xe x )  =  A )
873adant2 1026 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  E! x  e.  RR*  ( C xe x )  =  A )
9 oveq2 6296 . . . . 5  |-  ( x  =  B  ->  ( C xe x )  =  ( C xe B ) )
109eqeq1d 2452 . . . 4  |-  ( x  =  B  ->  (
( C xe x )  =  A  <-> 
( C xe B )  =  A ) )
1110riota2 6272 . . 3  |-  ( ( B  e.  RR*  /\  E! x  e.  RR*  ( C xe x )  =  A )  -> 
( ( C xe B )  =  A  <->  ( iota_ x  e. 
RR*  ( C xe x )  =  A )  =  B ) )
125, 8, 11syl2anc 666 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( ( C xe B )  =  A  <->  ( iota_ x  e. 
RR*  ( C xe x )  =  A )  =  B ) )
134, 12bitr4d 260 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  -> 
( ( A /𝑒  C )  =  B  <->  ( C xe B )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   E!wreu 2738   iota_crio 6249  (class class class)co 6288   RRcr 9535   0cc0 9536   RR*cxr 9671   xecxmu 11405   /𝑒 cxdiv 28379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-xneg 11406  df-xmul 11408  df-xdiv 28380
This theorem is referenced by:  xdivrec  28389
  Copyright terms: Public domain W3C validator