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Theorem enrer 9472
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 9463 . 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 9429 . 2  |-  ( x  +P.  y )  =  ( y  +P.  x
)
3 addclpr 9426 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  e.  P. )
4 addasspr 9430 . 2  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
5 addcanpr 9454 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  y )  =  ( x  +P.  z )  ->  y  =  z ) )
61, 2, 3, 4, 5ecopover 7452 1  |-  ~R  Er  ( P.  X.  P. )
Colors of variables: wff setvar class
Syntax hints:    X. cxp 4821    Er wer 7345   P.cnp 9267    +P. cpp 9269    ~R cer 9272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-omul 7172  df-er 7348  df-ni 9280  df-pli 9281  df-mi 9282  df-lti 9283  df-plpq 9316  df-mpq 9317  df-ltpq 9318  df-enq 9319  df-nq 9320  df-erq 9321  df-plq 9322  df-mq 9323  df-1nq 9324  df-rq 9325  df-ltnq 9326  df-np 9389  df-plp 9391  df-ltp 9393  df-enr 9463
This theorem is referenced by:  enreceq  9473  prsrlem1  9479  addsrmo  9480  mulsrmo  9481  ltsrpr  9484  0nsr  9486  axcnex  9554  wuncn  9577
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