Theorem enrbreq 3968

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 3960	. 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 3244	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 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  <.cop 1810   class class class wbr 2054  (class class class)co 3001  P.cnp 3779   +P. cpp 3781   ~R cer 3786

This theorem is referenced by:  enreceq 3971  addcmpblnr 3975  mulcmpblnr 3977

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-enr 3960

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