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Theorem nnssi2 24269
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 3151 . . . 4  |-  ( A  e.  NN  ->  A  e.  D )
31sseli 3151 . . . 4  |-  ( B  e.  NN  ->  B  e.  D )
4 nnssi2.2 . . . 4  |-  ( B  e.  NN  ->  ph )
52, 3, 43anim123i 1142 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  B  e.  NN )  ->  ( A  e.  D  /\  B  e.  D  /\  ph ) )
653anidm23 1246 . 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 17 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    e. wcel 1621    C_ wss 3127   NNcn 9714
This theorem is referenced by:  nndivsub  24271
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-in 3134  df-ss 3141
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