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Theorem dfcnqs 7617
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in R. The trick involves qsid 6462, 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 is a quotient set, even though it is not (compare df-c 7594), 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.)
Assertion
Ref Expression
dfcnqs ℂ = ((R × R) / E )

Proof of Theorem dfcnqs
StepHypRef Expression
1 df-c 7594 . 2 ℂ = (R × R)
2 qsid 6462 . 2 ((R × R) / E ) = (R × R)
31, 2eqtr4i 2141 1 ℂ = ((R × R) / E )
Colors of variables: wff set class
Syntax hints:   = wceq 1316   E cep 4179   × cxp 4507  ccnv 4508   / cqs 6396  Rcnr 7073  cc 7586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-eprel 4181  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399  df-qs 6403  df-c 7594
This theorem is referenced by:  axmulcom  7647  axaddass  7648  axmulass  7649  axdistr  7650
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