Theorem enqbreq 7687

Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.)

Assertion

Ref	Expression
enqbreq	|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. ~Q <. C , D >. <-> ( A .N D ) = ( B .N C ) ) )

Proof of Theorem enqbreq

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

Step	Hyp	Ref	Expression
1		df-enq 7678	. 2 |- ~Q = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) ) ) }
2	1	ecopoveq 6877	1 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. ~Q <. C , D >. <-> ( A .N D ) = ( B .N C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   <.cop 3697   class class class wbr 4114  (class class class)co 6058   N.cnpi 7603   .N cmi 7605   ~Q ceq 7610

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-iota 5317  df-fv 5365  df-ov 6061  df-enq 7678

This theorem is referenced by:  enqbreq2  7688  enqeceq  7690  enqdc  7692  addcmpblnq  7698  mulcmpblnq  7699  mulcanenq  7716