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Theorem enqer 8540
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. )
Dummy variables  x  y  z  w  v  u are mutually distinct and distinct from all other variables.

Proof of Theorem enqer
StepHypRef Expression
1 df-enq 8530 . 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 8515 . 2  |-  ( x  .N  y )  =  ( y  .N  x
)
3 mulclpi 8512 . 2  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( x  .N  y
)  e.  N. )
4 mulasspi 8516 . 2  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5 mulcanpi 8519 . . 3  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( ( x  .N  y )  =  ( x  .N  z )  <-> 
y  =  z ) )
65biimpd 200 . 2  |-  ( ( x  e.  N.  /\  y  e.  N. )  ->  ( ( x  .N  y )  =  ( x  .N  z )  ->  y  =  z ) )
71, 2, 3, 4, 6ecopover 6757 1  |-  ~Q  Er  ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1624    e. wcel 1685    X. cxp 4686  (class class class)co 5819    Er wer 6652   N.cnpi 8461    .N cmi 8463    ~Q ceq 8468
This theorem is referenced by:  nqereu  8548  nqerf  8549  nqerid  8552  enqeq  8553  nqereq  8554  adderpq  8575  mulerpq  8576  1nqenq  8581
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-mi 8493  df-enq 8530
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