MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfcnqs Structured version   Unicode version

Theorem dfcnqs 9017
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6970, 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 8996), 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 8996 . 2  |-  CC  =  ( R.  X.  R. )
2 qsid 6970 . 2  |-  ( ( R.  X.  R. ) /. `'  _E  )  =  ( R.  X.  R. )
31, 2eqtr4i 2459 1  |-  CC  =  ( ( R.  X.  R. ) /. `'  _E  )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    _E cep 4492    X. cxp 4876   `'ccnv 4877   /.cqs 6904   R.cnr 8742   CCcc 8988
This theorem is referenced by:  axmulcom  9030  axaddass  9031  axmulass  9032  axdistr  9033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-eprel 4494  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-ec 6907  df-qs 6911  df-c 8996
  Copyright terms: Public domain W3C validator