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Theorem mptelee 24981
Description: A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)
Assertion
Ref Expression
mptelee  |-  ( N  e.  NN  ->  (
( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  e.  ( EE `  N )  <->  A. k  e.  (
1 ... N ) ( A F B )  e.  RR ) )
Distinct variable group:    k, N
Allowed substitution hints:    A( k)    B( k)    F( k)

Proof of Theorem mptelee
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 elee 24980 . 2  |-  ( N  e.  NN  ->  (
( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  e.  ( EE `  N )  <-> 
( k  e.  ( 1 ... N ) 
|->  ( A F B ) ) : ( 1 ... N ) --> RR ) )
2 ovex 6348 . . . . 5  |-  ( A F B )  e. 
_V
3 eqid 2462 . . . . 5  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  =  ( k  e.  ( 1 ... N
)  |->  ( A F B ) )
42, 3fnmpti 5732 . . . 4  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  Fn  ( 1 ... N )
5 df-f 5609 . . . 4  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  ( ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  Fn  (
1 ... N )  /\  ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR ) )
64, 5mpbiran 934 . . 3  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR )
73rnmpt 5102 . . . . 5  |-  ran  (
k  e.  ( 1 ... N )  |->  ( A F B ) )  =  { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }
87sseq1i 3468 . . . 4  |-  ( ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR  <->  { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }  C_  RR )
9 abss 3510 . . . . 5  |-  ( { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }  C_  RR  <->  A. a
( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR ) )
10 nfre1 2860 . . . . . . . . 9  |-  F/ k E. k  e.  ( 1 ... N ) a  =  ( A F B )
11 nfv 1772 . . . . . . . . 9  |-  F/ k  a  e.  RR
1210, 11nfim 2014 . . . . . . . 8  |-  F/ k ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )
1312nfal 2041 . . . . . . 7  |-  F/ k A. a ( E. k  e.  ( 1 ... N ) a  =  ( A F B )  ->  a  e.  RR )
14 r19.23v 2879 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) ( a  =  ( A F B )  -> 
a  e.  RR )  <-> 
( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR ) )
1514albii 1702 . . . . . . . 8  |-  ( A. a A. k  e.  ( 1 ... N ) ( a  =  ( A F B )  ->  a  e.  RR ) 
<-> 
A. a ( E. k  e.  ( 1 ... N ) a  =  ( A F B )  ->  a  e.  RR ) )
16 ralcom4 3078 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) A. a ( a  =  ( A F B )  ->  a  e.  RR )  <->  A. a A. k  e.  ( 1 ... N
) ( a  =  ( A F B )  ->  a  e.  RR ) )
17 rsp 2766 . . . . . . . . . 10  |-  ( A. k  e.  ( 1 ... N ) A. a ( a  =  ( A F B )  ->  a  e.  RR )  ->  ( k  e.  ( 1 ... N )  ->  A. a
( a  =  ( A F B )  ->  a  e.  RR ) ) )
182clel2 3187 . . . . . . . . . 10  |-  ( ( A F B )  e.  RR  <->  A. a
( a  =  ( A F B )  ->  a  e.  RR ) )
1917, 18syl6ibr 235 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) A. a ( a  =  ( A F B )  ->  a  e.  RR )  ->  ( k  e.  ( 1 ... N )  ->  ( A F B )  e.  RR ) )
2016, 19sylbir 218 . . . . . . . 8  |-  ( A. a A. k  e.  ( 1 ... N ) ( a  =  ( A F B )  ->  a  e.  RR )  ->  ( k  e.  ( 1 ... N
)  ->  ( A F B )  e.  RR ) )
2115, 20sylbir 218 . . . . . . 7  |-  ( A. a ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )  ->  ( k  e.  ( 1 ... N
)  ->  ( A F B )  e.  RR ) )
2213, 21ralrimi 2800 . . . . . 6  |-  ( A. a ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )  ->  A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
23 nfra1 2781 . . . . . . . 8  |-  F/ k A. k  e.  ( 1 ... N ) ( A F B )  e.  RR
24 rsp 2766 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  (
k  e.  ( 1 ... N )  -> 
( A F B )  e.  RR ) )
25 eleq1a 2535 . . . . . . . . 9  |-  ( ( A F B )  e.  RR  ->  (
a  =  ( A F B )  -> 
a  e.  RR ) )
2624, 25syl6 34 . . . . . . . 8  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  (
k  e.  ( 1 ... N )  -> 
( a  =  ( A F B )  ->  a  e.  RR ) ) )
2723, 11, 26rexlimd 2883 . . . . . . 7  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  ( E. k  e.  (
1 ... N ) a  =  ( A F B )  ->  a  e.  RR ) )
2827alrimiv 1784 . . . . . 6  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  A. a
( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR ) )
2922, 28impbii 192 . . . . 5  |-  ( A. a ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR ) 
<-> 
A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
309, 29bitri 257 . . . 4  |-  ( { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }  C_  RR  <->  A. k  e.  ( 1 ... N
) ( A F B )  e.  RR )
318, 30bitri 257 . . 3  |-  ( ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR  <->  A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
326, 31bitri 257 . 2  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
331, 32syl6bb 269 1  |-  ( N  e.  NN  ->  (
( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  e.  ( EE `  N )  <->  A. k  e.  (
1 ... N ) ( A F B )  e.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1453    = wceq 1455    e. wcel 1898   {cab 2448   A.wral 2749   E.wrex 2750    C_ wss 3416    |-> cmpt 4477   ran crn 4857    Fn wfn 5600   -->wf 5601   ` cfv 5605  (class class class)co 6320   RRcr 9569   1c1 9571   NNcn 10642   ...cfz 11819   EEcee 24974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-map 7505  df-ee 24977
This theorem is referenced by:  eleesub  24997  eleesubd  24998  axsegconlem1  25003  axsegconlem8  25010  axpasch  25027  axeuclidlem  25048  axcontlem2  25051
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