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Theorem enrer 8907
Description: The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
Assertion
Ref Expression
enrer  |-  ~R  Er  ( P.  X.  P. )

Proof of Theorem enrer
Dummy variables  x  y  z  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-enr 8898 . 2  |-  ~R  =  { <. x ,  y
>.  |  ( (
x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }
2 addcompr 8862 . 2  |-  ( x  +P.  y )  =  ( y  +P.  x
)
3 addclpr 8859 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  e.  P. )
4 addasspr 8863 . 2  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
5 addcanpr 8887 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  y )  =  ( x  +P.  z )  ->  y  =  z ) )
61, 2, 3, 4, 5ecopover 6975 1  |-  ~R  Er  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    X. cxp 4843    Er wer 6869   P.cnp 8698    +P. cpp 8700    ~R cer 8705
This theorem is referenced by:  enreceq  8908  addsrpr  8914  mulsrpr  8915  ltsrpr  8916  0nsr  8918  axcnex  8986  wuncn  9009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6872  df-ni 8713  df-pli 8714  df-mi 8715  df-lti 8716  df-plpq 8749  df-mpq 8750  df-ltpq 8751  df-enq 8752  df-nq 8753  df-erq 8754  df-plq 8755  df-mq 8756  df-1nq 8757  df-rq 8758  df-ltnq 8759  df-np 8822  df-plp 8824  df-ltp 8826  df-enr 8898
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