Theorem nnssi3 10415

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 2068	. . 3 |- (A e. NN -> A e. D)
4	2	sseli 2068	. . 3 |- (B e. NN -> B e. D)
5	2	sseli 2068	. . 3 |- (C e. NN -> C e. D)
6	3, 4, 5	3anim123i 823	. 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 804	. 2 |- ((A e. NN /\ B e. NN /\ C e. NN) -> ph)
9	1, 6, 8	sylanc 473	1 |- ((A e. NN /\ B e. NN /\ C e. NN) -> ps)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   e. wcel 960   (_ wss 2050  NNcn 5308

This theorem is referenced by:  nndivsub 10416

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056

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