Users' Mathboxes Mathbox for Jeff Hoffman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nnssi2 Unicode version

Theorem nnssi2 24966
Description: Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
Hypotheses
Ref Expression
nnssi2.1  |-  NN  C_  D
nnssi2.2  |-  ( B  e.  NN  ->  ph )
nnssi2.3  |-  ( ( A  e.  D  /\  B  e.  D  /\  ph )  ->  ps )
Assertion
Ref Expression
nnssi2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ps )

Proof of Theorem nnssi2
StepHypRef Expression
1 nnssi2.1 . . . . 5  |-  NN  C_  D
21sseli 3189 . . . 4  |-  ( A  e.  NN  ->  A  e.  D )
31sseli 3189 . . . 4  |-  ( B  e.  NN  ->  B  e.  D )
4 nnssi2.2 . . . 4  |-  ( B  e.  NN  ->  ph )
52, 3, 43anim123i 1137 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  B  e.  NN )  ->  ( A  e.  D  /\  B  e.  D  /\  ph ) )
653anidm23 1241 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  e.  D  /\  B  e.  D  /\  ph ) )
7 nnssi2.3 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  ph )  ->  ps )
86, 7syl 15 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696    C_ wss 3165   NNcn 9762
This theorem is referenced by:  nndivsub  24968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
  Copyright terms: Public domain W3C validator