Theorem nnssi3 32455

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

Step	Hyp	Ref	Expression
1		nnssi3.1	. . . 4 |- NN C_ D
2	1	sseli 3599	. . 3 |- ( A e. NN -> A e. D )
3	1	sseli 3599	. . 3 |- ( B e. NN -> B e. D )
4	1	sseli 3599	. . 3 |- ( C e. NN -> C e. D )
5	2, 3, 4	3anim123i 1247	. 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 )
7	6	3ad2ant3 1084	. 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 )
9	5, 7, 8	syl2anc 693	1 |- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ps )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   C_ wss 3574   NNcn 11020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by: (None)