Theorem dfcnqs 11227

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 11206), 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 11206	. 2 ⊢ ℂ = (R × R)
2		qsid 8802	. 2 ⊢ ((R × R) / ◡ E ) = (R × R)
3	1, 2	eqtr4i 2787	1 ⊢ ℂ = ((R × R) / ◡ E )

This proof depends on syntax axioms:   = wceq 1570   E cep 5550   × cxp 5649  ^◡ccnv 5650   / cqs 8716  Rcnr 10950  ℂcc 11198

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-eprel 5551  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8719  df-qs 8723  df-c 11206

This theorem is used by:  axmulcom  11240  axaddass  11241  axmulass  11242  axdistr  11243