Theorem dfcnqs 6414

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 5360, 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 6392), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc.

Assertion

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

Proof of Theorem dfcnqs

Step	Hyp	Ref	Expression
1		df-c 6392	. 2 |- CC = (R. X. R.)
2		qsid 5360	. 2 |- ((R. X. R.)/.`' _E ) = (R. X. R.)
3	1, 2	eqtr4i 1911	1 |- CC = ((R. X. R.)/.`' _E )

Syntax hints:   = wceq 1298   _E cep 3581   X. cxp 3984  `'ccnv 3985  /.cqs 5317  R.cnr 6145  CCcc 6384

This theorem is referenced by:  axaddcom 6428  axmulcom 6429  axaddass 6430  axmulass 6431  axdistr 6432

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-eprel 3583  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-ec 5320  df-qs 5323  df-c 6392

Copyright terms: Public domain