Theorem dfcnqs 5234

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 4285, 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 5212), 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 5212	. 2 |- CC = (R. X. R.)
2		qsid 4285	. 2 |- ((R. X. R.)/.`'E) = (R. X. R.)
3	1, 2	eqtr4 1490	1 |- CC = ((R. X. R.)/.`'E)

Syntax hints:   = wceq 953  Ecep 2819   X. cxp 3158  `'ccnv 3159  /.cqs 4244  R.cnr 4965  CCcc 5204

This theorem is referenced by:  axaddcom 5247  axmulcom 5248  axaddass 5249  axmulass 5250  axdistr 5251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-eprel 2821  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-ec 4247  df-qs 4250  df-c 5212

Copyright terms: Public domain