Theorem dfcnqs 11211

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8841, 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 11190), 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 11190	. 2 ⊢ ℂ = (R × R)
2		qsid 8841	. 2 ⊢ ((R × R) / ◡ E ) = (R × R)
3	1, 2	eqtr4i 2771	1 ⊢ ℂ = ((R × R) / ◡ E )

Syntax hints:   = wceq 1537   E cep 5598   × cxp 5698  ^◡ccnv 5699   / cqs 8762  Rcnr 10934  ℂcc 11182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-qs 8769  df-c 11190

This theorem is referenced by:  axmulcom  11224  axaddass  11225  axmulass  11226  axdistr  11227