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Theorem nnssi2 26109
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 3304 . . . 4  |-  ( A  e.  NN  ->  A  e.  D )
31sseli 3304 . . . 4  |-  ( B  e.  NN  ->  B  e.  D )
4 nnssi2.2 . . . 4  |-  ( B  e.  NN  ->  ph )
52, 3, 43anim123i 1139 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  B  e.  NN )  ->  ( A  e.  D  /\  B  e.  D  /\  ph ) )
653anidm23 1243 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721    C_ wss 3280   NNcn 9956
This theorem is referenced by:  nndivsub  26111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-in 3287  df-ss 3294
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