Theorem dfcnqs 9963

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 7813, 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 9942), 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.) (New usage is discouraged.)

Assertion

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

Proof of Theorem dfcnqs

Step	Hyp	Ref	Expression
1		df-c 9942	. 2 |- CC = ( R. X. R. )
2		qsid 7813	. 2 |- ( ( R. X. R. ) /. `' _E ) = ( R. X. R. )
3	1, 2	eqtr4i 2647	1 |- CC = ( ( R. X. R. ) /. `' _E )

Syntax hints:   = wceq 1483   _E cep 5028   X. cxp 5112   `'ccnv 5113   /.cqs 7741   R.cnr 9687   CCcc 9934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748  df-c 9942

This theorem is referenced by:  axmulcom  9976  axaddass  9977  axmulass  9978  axdistr  9979