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Theorem elee 23152
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 6111 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21oveq2d 6119 . . . 4  |-  ( n  =  N  ->  ( RR  ^m  ( 1 ... n ) )  =  ( RR  ^m  (
1 ... N ) ) )
3 df-ee 23149 . . . 4  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
4 ovex 6128 . . . 4  |-  ( RR 
^m  ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5786 . . 3  |-  ( N  e.  NN  ->  ( EE `  N )  =  ( RR  ^m  (
1 ... N ) ) )
65eleq2d 2510 . 2  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( RR  ^m  (
1 ... N ) ) ) )
7 reex 9385 . . 3  |-  RR  e.  _V
8 ovex 6128 . . 3  |-  ( 1 ... N )  e. 
_V
97, 8elmap 7253 . 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 1369    e. wcel 1756   -->wf 5426   ` cfv 5430  (class class class)co 6103    ^m cmap 7226   RRcr 9293   1c1 9295   NNcn 10334   ...cfz 11449   EEcee 23146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-map 7228  df-ee 23149
This theorem is referenced by:  mptelee  23153  eleei  23155  axlowdimlem5  23204  axlowdimlem7  23206  axlowdimlem10  23209  axlowdimlem14  23213  axlowdim1  23217
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