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

Theorem dfcnqs 9842
 Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 7700, 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 ℂ is a quotient set, even though it is not (compare df-c 9821), 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 ℂ = ((R × R) / E )

Proof of Theorem dfcnqs
StepHypRef Expression
1 df-c 9821 . 2 ℂ = (R × R)
2 qsid 7700 . 2 ((R × R) / E ) = (R × R)
31, 2eqtr4i 2635 1 ℂ = ((R × R) / E )
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   E cep 4947   × cxp 5036  ◡ccnv 5037   / cqs 7628  Rcnr 9566  ℂcc 9813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631  df-qs 7635  df-c 9821 This theorem is referenced by:  axmulcom  9855  axaddass  9856  axmulass  9857  axdistr  9858
 Copyright terms: Public domain W3C validator