Theorem dfcnqs 11161

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8802, 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 11140), 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 11140	. 2 ⊢ ℂ = (R × R)
2		qsid 8802	. 2 ⊢ ((R × R) / ◡ E ) = (R × R)
3	1, 2	eqtr4i 2762	1 ⊢ ℂ = ((R × R) / ◡ E )

Syntax hints:   = wceq 1540   E cep 5557   × cxp 5657  ^◡ccnv 5658   / cqs 8723  Rcnr 10884  ℂcc 11132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-eprel 5558  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8726  df-qs 8730  df-c 11140

This theorem is referenced by:  axmulcom  11174  axaddass  11175  axmulass  11176  axdistr  11177