Theorem enrbreq 5174

Description: Equivalence relation for signed reals in terms of positive reals.

Assertion

Ref	Expression
enrbreq	|- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> (<.A, B>. ~R <.C, D>. <-> (A +P. D) = (B +P. C)))

Proof of Theorem enrbreq

Step	Hyp	Ref	Expression
1		df-enr 5166	. 2 |- ~R = {<.x, y>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (z +P. u) = (w +P. v)))}
2	1	ecopopreq 4308	1 |- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> (<.A, B>. ~R <.C, D>. <-> (A +P. D) = (B +P. C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  <.cop 2411   class class class wbr 2619  (class class class)co 3963  P.cnp 4985   +P. cpp 4987   ~R cer 4992

This theorem is referenced by:  enreceq 5177  addcmpblnr 5181  mulcmpblnr 5183

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-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-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-enr 5166

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