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Theorem dfcnqs 9565
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 7437, 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 9544), 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.) (New usage is discouraged.)
Assertion
Ref Expression
dfcnqs  |-  CC  =  ( ( R.  X.  R. ) /. `'  _E  )

Proof of Theorem dfcnqs
StepHypRef Expression
1 df-c 9544 . 2  |-  CC  =  ( R.  X.  R. )
2 qsid 7437 . 2  |-  ( ( R.  X.  R. ) /. `'  _E  )  =  ( R.  X.  R. )
31, 2eqtr4i 2461 1  |-  CC  =  ( ( R.  X.  R. ) /. `'  _E  )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    _E cep 4763    X. cxp 4852   `'ccnv 4853   /.cqs 7370   R.cnr 9289   CCcc 9536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-eprel 4765  df-xp 4860  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ec 7373  df-qs 7377  df-c 9544
This theorem is referenced by:  axmulcom  9578  axaddass  9579  axmulass  9580  axdistr  9581
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