Theorem dfcnqs 11126

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 8778, 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 11105), 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 11105	. 2 ⊢ ℂ = (R × R)
2		qsid 8778	. 2 ⊢ ((R × R) / ◡ E ) = (R × R)
3	1, 2	eqtr4i 2795	1 ⊢ ℂ = ((R × R) / ◡ E )

Syntax hints:   = wceq 1567   E cep 5561   × cxp 5660  ^◡ccnv 5661   / cqs 8692  Rcnr 10849  ℂcc 11097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-eprel 5562  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ec 8695  df-qs 8699  df-c 11105

This theorem is referenced by:  axmulcom  11139  axaddass  11140  axmulass  11141  axdistr  11142