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Theorem elee 24323
Description: Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over 
N space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
elee  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A :
( 1 ... N
) --> RR ) )

Proof of Theorem elee
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 oveq2 6304 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21oveq2d 6312 . . . 4  |-  ( n  =  N  ->  ( RR  ^m  ( 1 ... n ) )  =  ( RR  ^m  (
1 ... N ) ) )
3 df-ee 24320 . . . 4  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
4 ovex 6324 . . . 4  |-  ( RR 
^m  ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5956 . . 3  |-  ( N  e.  NN  ->  ( EE `  N )  =  ( RR  ^m  (
1 ... N ) ) )
65eleq2d 2527 . 2  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( RR  ^m  (
1 ... N ) ) ) )
7 reex 9600 . . 3  |-  RR  e.  _V
8 ovex 6324 . . 3  |-  ( 1 ... N )  e. 
_V
97, 8elmap 7466 . 2  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> RR )
106, 9syl6bb 261 1  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A :
( 1 ... N
) --> RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   RRcr 9508   1c1 9510   NNcn 10556   ...cfz 11697   EEcee 24317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-ee 24320
This theorem is referenced by:  mptelee  24324  eleei  24326  axlowdimlem5  24375  axlowdimlem7  24377  axlowdimlem10  24380  axlowdimlem14  24384  axlowdim1  24388
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