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Theorem enrer 8692
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 8683 . 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 8647 . 2  |-  ( x  +P.  y )  =  ( y  +P.  x
)
3 addclpr 8644 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  e.  P. )
4 addasspr 8648 . 2  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
5 addcanpr 8672 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  y )  =  ( x  +P.  z )  ->  y  =  z ) )
61, 2, 3, 4, 5ecopover 6764 1  |-  ~R  Er  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    X. cxp 4689    Er wer 6659   P.cnp 8483    +P. cpp 8485    ~R cer 8490
This theorem is referenced by:  enreceq  8693  addsrpr  8699  mulsrpr  8700  ltsrpr  8701  0nsr  8703  axcnex  8771  wuncn  8794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-ni 8498  df-pli 8499  df-mi 8500  df-lti 8501  df-plpq 8534  df-mpq 8535  df-ltpq 8536  df-enq 8537  df-nq 8538  df-erq 8539  df-plq 8540  df-mq 8541  df-1nq 8542  df-rq 8543  df-ltnq 8544  df-np 8607  df-plp 8609  df-ltp 8611  df-enr 8683
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