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Theorem elee 24594
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 5882 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21oveq2d 5890 . . . 4  |-  ( n  =  N  ->  ( RR  ^m  ( 1 ... n ) )  =  ( RR  ^m  (
1 ... N ) ) )
3 df-ee 24591 . . . 4  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
4 ovex 5899 . . . 4  |-  ( RR 
^m  ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5618 . . 3  |-  ( N  e.  NN  ->  ( EE `  N )  =  ( RR  ^m  (
1 ... N ) ) )
65eleq2d 2363 . 2  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( RR  ^m  (
1 ... N ) ) ) )
7 reex 8844 . . 3  |-  RR  e.  _V
8 ovex 5899 . . 3  |-  ( 1 ... N )  e. 
_V
97, 8elmap 6812 . 2  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> RR )
106, 9syl6bb 252 1  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A :
( 1 ... N
) --> RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   RRcr 8752   1c1 8754   NNcn 9762   ...cfz 10798   EEcee 24588
This theorem is referenced by:  mptelee  24595  eleei  24597  axlowdimlem5  24646  axlowdimlem7  24648  axlowdimlem10  24651  axlowdimlem14  24655  axlowdim1  24659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-ee 24591
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