Theorem dfcnqs 7858

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6618, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that CC is a quotient set, even though it is not (compare df-c 7835), and allows us 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.)

Assertion

Ref	Expression
dfcnqs	|- CC = ( ( R. X. R. ) /. `' _E )

Proof of Theorem dfcnqs

Step	Hyp	Ref	Expression
1		df-c 7835	. 2 |- CC = ( R. X. R. )
2		qsid 6618	. 2 |- ( ( R. X. R. ) /. `' _E ) = ( R. X. R. )
3	1, 2	eqtr4i 2213	1 |- CC = ( ( R. X. R. ) /. `' _E )

Syntax hints:   = wceq 1364   _E cep 4302   X. cxp 4639   `'ccnv 4640   /.cqs 6552   R.cnr 7314   CCcc 7827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-eprel 4304  df-xp 4647  df-cnv 4649  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-ec 6555  df-qs 6559  df-c 7835

This theorem is referenced by:  axmulcom  7888  axaddass  7889  axmulass  7890  axdistr  7891