Theorem dfcnqs 11033

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8705, 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 ℂ is a quotient set, even though it is not (compare df-c 11012), and allows 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

Step	Hyp	Ref	Expression
1		df-c 11012	. 2 ⊢ ℂ = (R × R)
2		qsid 8705	. 2 ⊢ ((R × R) / ◡ E ) = (R × R)
3	1, 2	eqtr4i 2757	1 ⊢ ℂ = ((R × R) / ◡ E )

Syntax hints:   = wceq 1541   E cep 5513   × cxp 5612  ^◡ccnv 5613   / cqs 8621  Rcnr 10756  ℂcc 11004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-eprel 5514  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624  df-qs 8628  df-c 11012

This theorem is referenced by:  axmulcom  11046  axaddass  11047  axmulass  11048  axdistr  11049