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Theorem enqer 8798
Description: The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
Assertion
Ref Expression
enqer  |-  ~Q  Er  ( N.  X.  N. )

Proof of Theorem enqer
Dummy variables  x  y  z  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-enq 8788 . 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 ) ) ) }
2 mulcompi 8773 . 2  |-  ( x  .N  y )  =  ( y  .N  x
)
3 mulclpi 8770 . 2  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( x  .N  y
)  e.  N. )
4 mulasspi 8774 . 2  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5 mulcanpi 8777 . . 3  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( ( x  .N  y )  =  ( x  .N  z )  <-> 
y  =  z ) )
65biimpd 199 . 2  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( ( x  .N  y )  =  ( x  .N  z )  ->  y  =  z ) )
71, 2, 3, 4, 6ecopover 7008 1  |-  ~Q  Er  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725    X. cxp 4876  (class class class)co 6081    Er wer 6902   N.cnpi 8719    .N cmi 8721    ~Q ceq 8726
This theorem is referenced by:  nqereu  8806  nqerf  8807  nqerid  8810  enqeq  8811  nqereq  8812  adderpq  8833  mulerpq  8834  1nqenq  8839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-mi 8751  df-enq 8788
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