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Theorem dfcnqs 10552
Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in R. The trick involves qsid 8352, which shows that the coset of the converse membership 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 10531), 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 10531 . 2 ℂ = (R × R)
2 qsid 8352 . 2 ((R × R) / E ) = (R × R)
31, 2eqtr4i 2844 1 ℂ = ((R × R) / E )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528   E cep 5457   × cxp 5546  ccnv 5547   / cqs 8277  Rcnr 10275  cc 10523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-eprel 5458  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280  df-qs 8284  df-c 10531
This theorem is referenced by:  axmulcom  10565  axaddass  10566  axmulass  10567  axdistr  10568
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