Theorem enrbreq 7882

Description: Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)

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

Dummy variables x y z w v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-enr 7874	. 2 |- ~R = { <. x , y >. | ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) ) }
2	1	ecopoveq 6740	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 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   <.cop 3646   class class class wbr 4059  (class class class)co 5967   P.cnp 7439   +P. cpp 7441   ~R cer 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-iota 5251  df-fv 5298  df-ov 5970  df-enr 7874

This theorem is referenced by:  enreceq  7884  addcmpblnr  7887  mulcmpblnr  7889