Theorem dfcnqs 5262

Description: Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 4301, 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 5240), 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 5240	. 2 |- CC = (R. X. R.)
2		qsid 4301	. 2 |- ((R. X. R.)/.`'E) = (R. X. R.)
3	1, 2	eqtr4 1498	1 |- CC = ((R. X. R.)/.`'E)

Syntax hints:   = wceq 956  Ecep 2830   X. cxp 3168  `'ccnv 3169  /.cqs 4260  R.cnr 4993  CCcc 5232

This theorem is referenced by:  axaddcom 5275  axmulcom 5276  axaddass 5277  axmulass 5278  axdistr 5279

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-eprel 2832  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-ec 4263  df-qs 4266  df-c 5240

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