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

Theorem dfcnqs 9022
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6973, 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 9001), 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 9001 . 2  |-  CC  =  ( R.  X.  R. )
2 qsid 6973 . 2  |-  ( ( R.  X.  R. ) /. `'  _E  )  =  ( R.  X.  R. )
31, 2eqtr4i 2461 1  |-  CC  =  ( ( R.  X.  R. ) /. `'  _E  )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    _E cep 4495    X. cxp 4879   `'ccnv 4880   /.cqs 6907   R.cnr 8747   CCcc 8993
This theorem is referenced by:  axmulcom  9035  axaddass  9036  axmulass  9037  axdistr  9038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-eprel 4497  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-ec 6910  df-qs 6914  df-c 9001
  Copyright terms: Public domain W3C validator