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Theorem enqer 8545
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 8535 . 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 8520 . 2  |-  ( x  .N  y )  =  ( y  .N  x
)
3 mulclpi 8517 . 2  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( x  .N  y
)  e.  N. )
4 mulasspi 8521 . 2  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5 mulcanpi 8524 . . 3  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( ( x  .N  y )  =  ( x  .N  z )  <-> 
y  =  z ) )
65biimpd 198 . 2  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( ( x  .N  y )  =  ( x  .N  z )  ->  y  =  z ) )
71, 2, 3, 4, 6ecopover 6762 1  |-  ~Q  Er  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    X. cxp 4687  (class class class)co 5858    Er wer 6657   N.cnpi 8466    .N cmi 8468    ~Q ceq 8473
This theorem is referenced by:  nqereu  8553  nqerf  8554  nqerid  8557  enqeq  8558  nqereq  8559  adderpq  8580  mulerpq  8581  1nqenq  8586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-mi 8498  df-enq 8535
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