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Theorem eleenn 22965
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 3630 . 2  |-  ( A  e.  ( EE `  N )  ->  -.  ( EE `  N )  =  (/) )
2 ovex 6105 . . . . 5  |-  ( RR 
^m  ( 1 ... n ) )  e. 
_V
3 df-ee 22960 . . . . 5  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
42, 3dmmpti 5528 . . . 4  |-  dom  EE  =  NN
54eleq2i 2497 . . 3  |-  ( N  e.  dom  EE  <->  N  e.  NN )
6 ndmfv 5702 . . 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 1362    e. wcel 1755   (/)c0 3625   dom cdm 4827   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   RRcr 9269   1c1 9271   NNcn 10310   ...cfz 11424   EEcee 22957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-iota 5369  df-fun 5408  df-fn 5409  df-fv 5414  df-ov 6083  df-ee 22960
This theorem is referenced by:  eleei  22966  eedimeq  22967  brbtwn  22968  brcgr  22969  eleesub  22980  eleesubd  22981  axsegconlem1  22986  axsegconlem8  22993  axeuclidlem  23031  brsegle  27986
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