Theorem dfcnqs 6917

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 6171, 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 6895), 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 6895	. 2 |- CC = ( R. X. R. )
2		qsid 6171	. 2 |- ( ( R. X. R. ) /. `' _E ) = ( R. X. R. )
3	1, 2	eqtr4i 2063	1 |- CC = ( ( R. X. R. ) /. `' _E )

Syntax hints:   = wceq 1243   _E cep 4024   X. cxp 4343   `'ccnv 4344   /.cqs 6105   R.cnr 6395   CCcc 6887

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-ec 6108  df-qs 6112  df-c 6895

This theorem is referenced by:  axmulcom  6945  axaddass  6946  axmulass  6947  axdistr  6948