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Theorem eleenn 24064
Description: If  A is in  ( EE `  N ), then 
N is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)
Assertion
Ref Expression
eleenn  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )

Proof of Theorem eleenn
StepHypRef Expression
1 n0i 3772 . 2  |-  ( A  e.  ( EE `  N )  ->  -.  ( EE `  N )  =  (/) )
2 ovex 6305 . . . . 5  |-  ( RR 
^m  ( 1 ... n ) )  e. 
_V
3 df-ee 24059 . . . . 5  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
42, 3dmmpti 5696 . . . 4  |-  dom  EE  =  NN
54eleq2i 2519 . . 3  |-  ( N  e.  dom  EE  <->  N  e.  NN )
6 ndmfv 5876 . . 3  |-  ( -.  N  e.  dom  EE  ->  ( EE `  N
)  =  (/) )
75, 6sylnbir 307 . 2  |-  ( -.  N  e.  NN  ->  ( EE `  N )  =  (/) )
81, 7nsyl2 127 1  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802   (/)c0 3767   dom cdm 4985   ` cfv 5574  (class class class)co 6277    ^m cmap 7418   RRcr 9489   1c1 9491   NNcn 10537   ...cfz 11676   EEcee 24056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-iota 5537  df-fun 5576  df-fn 5577  df-fv 5582  df-ov 6280  df-ee 24059
This theorem is referenced by:  eleei  24065  eedimeq  24066  brbtwn  24067  brcgr  24068  eleesub  24079  eleesubd  24080  axsegconlem1  24085  axsegconlem8  24092  axeuclidlem  24130  brsegle  29726
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