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Theorem dfcnqs 10252
 Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8052, 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 10231), 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 10231 . 2 ℂ = (R × R)
2 qsid 8052 . 2 ((R × R) / E ) = (R × R)
31, 2eqtr4i 2825 1 ℂ = ((R × R) / E )
 Colors of variables: wff setvar class Syntax hints:   = wceq 1653   E cep 5225   × cxp 5311  ◡ccnv 5312   / cqs 7982  Rcnr 9976  ℂcc 10223 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-eprel 5226  df-xp 5319  df-cnv 5321  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-ec 7985  df-qs 7989  df-c 10231 This theorem is referenced by:  axmulcom  10265  axaddass  10266  axmulass  10267  axdistr  10268
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