Theorem dfcnqs 8104

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6812, 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 8081), 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 8081	. 2 |- CC = ( R. X. R. )
2		qsid 6812	. 2 |- ( ( R. X. R. ) /. `' _E ) = ( R. X. R. )
3	1, 2	eqtr4i 2255	1 |- CC = ( ( R. X. R. ) /. `' _E )

Syntax hints:   = wceq 1398   _E cep 4390   X. cxp 4729   `'ccnv 4730   /.cqs 6744   R.cnr 7560   CCcc 8073

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-eprel 4392  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-ec 6747  df-qs 6751  df-c 8081

This theorem is referenced by:  axmulcom  8134  axaddass  8135  axmulass  8136  axdistr  8137