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Theorem elee 24910
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 6309 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21oveq2d 6317 . . . 4  |-  ( n  =  N  ->  ( RR  ^m  ( 1 ... n ) )  =  ( RR  ^m  (
1 ... N ) ) )
3 df-ee 24907 . . . 4  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
4 ovex 6329 . . . 4  |-  ( RR 
^m  ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5960 . . 3  |-  ( N  e.  NN  ->  ( EE `  N )  =  ( RR  ^m  (
1 ... N ) ) )
65eleq2d 2492 . 2  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( RR  ^m  (
1 ... N ) ) ) )
7 reex 9630 . . 3  |-  RR  e.  _V
8 ovex 6329 . . 3  |-  ( 1 ... N )  e. 
_V
97, 8elmap 7504 . 2  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> RR )
106, 9syl6bb 264 1  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A :
( 1 ... N
) --> RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1868   -->wf 5593   ` cfv 5597  (class class class)co 6301    ^m cmap 7476   RRcr 9538   1c1 9540   NNcn 10609   ...cfz 11784   EEcee 24904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-ee 24907
This theorem is referenced by:  mptelee  24911  eleei  24913  axlowdimlem5  24962  axlowdimlem7  24964  axlowdimlem10  24967  axlowdimlem14  24971  axlowdim1  24975
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