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Theorem nnssi3 29498
Description: Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
Hypotheses
Ref Expression
nnssi3.1  |-  NN  C_  D
nnssi3.2  |-  ( C  e.  NN  ->  ph )
nnssi3.3  |-  ( ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D
)  /\  ph )  ->  ps )
Assertion
Ref Expression
nnssi3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ps )

Proof of Theorem nnssi3
StepHypRef Expression
1 nnssi3.1 . . . 4  |-  NN  C_  D
21sseli 3500 . . 3  |-  ( A  e.  NN  ->  A  e.  D )
31sseli 3500 . . 3  |-  ( B  e.  NN  ->  B  e.  D )
41sseli 3500 . . 3  |-  ( C  e.  NN  ->  C  e.  D )
52, 3, 43anim123i 1181 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( A  e.  D  /\  B  e.  D  /\  C  e.  D )
)
6 nnssi3.2 . . 3  |-  ( C  e.  NN  ->  ph )
763ad2ant3 1019 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ph )
8 nnssi3.3 . 2  |-  ( ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D
)  /\  ph )  ->  ps )
95, 7, 8syl2anc 661 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767    C_ wss 3476   NNcn 10532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490
This theorem is referenced by: (None)
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