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

Theorem enqbreq 7305
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 7296 . 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 6604 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 1348    e. wcel 2141   <.cop 3584   class class class wbr 3987  (class class class)co 5850   N.cnpi 7221    .N cmi 7223    ~Q ceq 7228
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-iota 5158  df-fv 5204  df-ov 5853  df-enq 7296
This theorem is referenced by:  enqbreq2  7306  enqeceq  7308  enqdc  7310  addcmpblnq  7316  mulcmpblnq  7317  mulcanenq  7334
  Copyright terms: Public domain W3C validator