Theorem dfcnqs 7956

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6689, 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 7933), 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 7933	. 2 |- CC = ( R. X. R. )
2		qsid 6689	. 2 |- ( ( R. X. R. ) /. `' _E ) = ( R. X. R. )
3	1, 2	eqtr4i 2229	1 |- CC = ( ( R. X. R. ) /. `' _E )

Syntax hints:   = wceq 1373   _E cep 4335   X. cxp 4674   `'ccnv 4675   /.cqs 6621   R.cnr 7412   CCcc 7925

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-eprel 4337  df-xp 4682  df-cnv 4684  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-ec 6624  df-qs 6628  df-c 7933

This theorem is referenced by:  axmulcom  7986  axaddass  7987  axmulass  7988  axdistr  7989