Theorem nnssi3 10413

Description: Convert a theorem for real/complex numbers into one for natural numbers.

Hypotheses

Ref	Expression
nnssi3.1	|- NN (_ 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		nnssi3.3	. 2 |- (((A e. D /\ B e. D /\ C e. D) /\ ph) -> ps)
2		nnssi3.1	. . . 4 |- NN (_ D
3	2	sseli 2063	. . 3 |- (A e. NN -> A e. D)
4	2	sseli 2063	. . 3 |- (B e. NN -> B e. D)
5	2	sseli 2063	. . 3 |- (C e. NN -> C e. D)
6	3, 4, 5	3anim123i 820	. 2 |- ((A e. NN /\ B e. NN /\ C e. NN) -> (A e. D /\ B e. D /\ C e. D))
7		nnssi3.2	. . 3 |- (C e. NN -> ph)
8	7	3ad2ant3 801	. 2 |- ((A e. NN /\ B e. NN /\ C e. NN) -> ph)
9	1, 6, 8	sylanc 471	1 |- ((A e. NN /\ B e. NN /\ C e. NN) -> ps)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   e. wcel 957   (_ wss 2045  NNcn 5283

This theorem is referenced by:  nndivsub 10414

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2049  df-ss 2051

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