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Theorem dfcnqs 6975
 Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 6202, 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 6953), 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 6953 . 2 ℂ = (R × R)
2 qsid 6202 . 2 ((R × R) / E ) = (R × R)
31, 2eqtr4i 2079 1 ℂ = ((R × R) / E )
 Colors of variables: wff set class Syntax hints:   = wceq 1259   E cep 4052   × cxp 4371  ◡ccnv 4372   / cqs 6136  Rcnr 6453  ℂcc 6945 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-eprel 4054  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-ec 6139  df-qs 6143  df-c 6953 This theorem is referenced by:  axmulcom  7003  axaddass  7004  axmulass  7005  axdistr  7006
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