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Theorem enrer 9438
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 9429 . 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 9395 . 2  |-  ( x  +P.  y )  =  ( y  +P.  x
)
3 addclpr 9392 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  e.  P. )
4 addasspr 9396 . 2  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
5 addcanpr 9420 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  y )  =  ( x  +P.  z )  ->  y  =  z ) )
61, 2, 3, 4, 5ecopover 7412 1  |-  ~R  Er  ( P.  X.  P. )
Colors of variables: wff setvar class
Syntax hints:    X. cxp 4997    Er wer 7305   P.cnp 9233    +P. cpp 9235    ~R cer 9238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ni 9246  df-pli 9247  df-mi 9248  df-lti 9249  df-plpq 9282  df-mpq 9283  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-plq 9288  df-mq 9289  df-1nq 9290  df-rq 9291  df-ltnq 9292  df-np 9355  df-plp 9357  df-ltp 9359  df-enr 9429
This theorem is referenced by:  enreceq  9439  prsrlem1  9445  addsrmo  9446  mulsrmo  9447  ltsrpr  9450  0nsr  9452  axcnex  9520  wuncn  9543
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