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Theorem elee 25825
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 6081 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21oveq2d 6089 . . . 4  |-  ( n  =  N  ->  ( RR  ^m  ( 1 ... n ) )  =  ( RR  ^m  (
1 ... N ) ) )
3 df-ee 25822 . . . 4  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
4 ovex 6098 . . . 4  |-  ( RR 
^m  ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5798 . . 3  |-  ( N  e.  NN  ->  ( EE `  N )  =  ( RR  ^m  (
1 ... N ) ) )
65eleq2d 2502 . 2  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( RR  ^m  (
1 ... N ) ) ) )
7 reex 9073 . . 3  |-  RR  e.  _V
8 ovex 6098 . . 3  |-  ( 1 ... N )  e. 
_V
97, 8elmap 7034 . 2  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> RR )
106, 9syl6bb 253 1  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A :
( 1 ... N
) --> RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   RRcr 8981   1c1 8983   NNcn 9992   ...cfz 11035   EEcee 25819
This theorem is referenced by:  mptelee  25826  eleei  25828  axlowdimlem5  25877  axlowdimlem7  25879  axlowdimlem10  25882  axlowdimlem14  25886  axlowdim1  25890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-ee 25822
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