ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  enqbreq Unicode version

Theorem enqbreq 7171
Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.)
Assertion
Ref Expression
enqbreq  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  ~Q  <. C ,  D >.  <->  ( A  .N  D )  =  ( B  .N  C ) ) )

Proof of Theorem enqbreq
Dummy variables  x  y  z  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-enq 7162 . 2  |-  ~Q  =  { <. x ,  y
>.  |  ( (
x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) ) ) }
21ecopoveq 6524 1  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  ~Q  <. C ,  D >.  <->  ( A  .N  D )  =  ( B  .N  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   <.cop 3530   class class class wbr 3929  (class class class)co 5774   N.cnpi 7087    .N cmi 7089    ~Q ceq 7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-iota 5088  df-fv 5131  df-ov 5777  df-enq 7162
This theorem is referenced by:  enqbreq2  7172  enqeceq  7174  enqdc  7176  addcmpblnq  7182  mulcmpblnq  7183  mulcanenq  7200
  Copyright terms: Public domain W3C validator