Theorem enrbreq 5186

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 5178	. 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 4314	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 958   e. wcel 960  <.cop 2415   class class class wbr 2624  (class class class)co 3969  P.cnp 4997   +P. cpp 4999   ~R cer 5004

This theorem is referenced by:  enreceq 5189  addcmpblnr 5193  mulcmpblnr 5195

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-enr 5178

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