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Theorem enrer 8686
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 8677 . 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 8641 . 2  |-  ( x  +P.  y )  =  ( y  +P.  x
)
3 addclpr 8638 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  e.  P. )
4 addasspr 8642 . 2  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
5 addcanpr 8666 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  y )  =  ( x  +P.  z )  ->  y  =  z ) )
61, 2, 3, 4, 5ecopover 6758 1  |-  ~R  Er  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    X. cxp 4686    Er wer 6653   P.cnp 8477    +P. cpp 8479    ~R cer 8484
This theorem is referenced by:  enreceq  8687  addsrpr  8693  mulsrpr  8694  ltsrpr  8695  0nsr  8697  axcnex  8765  wuncn  8788
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338
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 1631  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-rmo 2552  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-int 3864  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 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-omul 6480  df-er 6656  df-ni 8492  df-pli 8493  df-mi 8494  df-lti 8495  df-plpq 8528  df-mpq 8529  df-ltpq 8530  df-enq 8531  df-nq 8532  df-erq 8533  df-plq 8534  df-mq 8535  df-1nq 8536  df-rq 8537  df-ltnq 8538  df-np 8601  df-plp 8603  df-ltp 8605  df-enr 8677
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