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Theorem dfcnqs 7613
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6460, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that  CC is a quotient set, even though it is not (compare df-c 7590), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.)
Assertion
Ref Expression
dfcnqs  |-  CC  =  ( ( R.  X.  R. ) /. `'  _E  )

Proof of Theorem dfcnqs
StepHypRef Expression
1 df-c 7590 . 2  |-  CC  =  ( R.  X.  R. )
2 qsid 6460 . 2  |-  ( ( R.  X.  R. ) /. `'  _E  )  =  ( R.  X.  R. )
31, 2eqtr4i 2139 1  |-  CC  =  ( ( R.  X.  R. ) /. `'  _E  )
Colors of variables: wff set class
Syntax hints:    = wceq 1314    _E cep 4177    X. cxp 4505   `'ccnv 4506   /.cqs 6394   R.cnr 7069   CCcc 7582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-eprel 4179  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-ec 6397  df-qs 6401  df-c 7590
This theorem is referenced by:  axmulcom  7643  axaddass  7644  axmulass  7645  axdistr  7646
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