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Theorem nnssi2 29888
Description: Convert a theorem for real/complex numbers into one for positive integers. (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 3482 . . . 4  |-  ( A  e.  NN  ->  A  e.  D )
31sseli 3482 . . . 4  |-  ( B  e.  NN  ->  B  e.  D )
4 nnssi2.2 . . . 4  |-  ( B  e.  NN  ->  ph )
52, 3, 43anim123i 1180 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  B  e.  NN )  ->  ( A  e.  D  /\  B  e.  D  /\  ph ) )
653anidm23 1286 . 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 16 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    e. wcel 1802    C_ wss 3458   NNcn 10537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-in 3465  df-ss 3472
This theorem is referenced by:  nndivsub  29890
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