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

Theorem dfcnqs 9009
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6962, 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 8988), 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 8988 . 2  |-  CC  =  ( R.  X.  R. )
2 qsid 6962 . 2  |-  ( ( R.  X.  R. ) /. `'  _E  )  =  ( R.  X.  R. )
31, 2eqtr4i 2458 1  |-  CC  =  ( ( R.  X.  R. ) /. `'  _E  )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    _E cep 4484    X. cxp 4868   `'ccnv 4869   /.cqs 6896   R.cnr 8734   CCcc 8980
This theorem is referenced by:  axmulcom  9022  axaddass  9023  axmulass  9024  axdistr  9025
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-eprel 4486  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-ec 6899  df-qs 6903  df-c 8988
  Copyright terms: Public domain W3C validator