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Theorem mptelee 23076
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 23075 . 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 6115 . . . . 5  |-  ( A F B )  e. 
_V
3 eqid 2441 . . . . 5  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  =  ( k  e.  ( 1 ... N
)  |->  ( A F B ) )
42, 3fnmpti 5536 . . . 4  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  Fn  ( 1 ... N )
5 df-f 5419 . . . 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 904 . . 3  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR )
73rnmpt 5081 . . . . 5  |-  ran  (
k  e.  ( 1 ... N )  |->  ( A F B ) )  =  { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }
87sseq1i 3377 . . . 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 3418 . . . . 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 2770 . . . . . . . . 9  |-  F/ k E. k  e.  ( 1 ... N ) a  =  ( A F B )
11 nfv 1678 . . . . . . . . 9  |-  F/ k  a  e.  RR
1210, 11nfim 1857 . . . . . . . 8  |-  F/ k ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )
1312nfal 1877 . . . . . . 7  |-  F/ k A. a ( E. k  e.  ( 1 ... N ) a  =  ( A F B )  ->  a  e.  RR )
14 r19.23v 2831 . . . . . . . . 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 1615 . . . . . . . 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 2989 . . . . . . . . 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 2774 . . . . . . . . . 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 3093 . . . . . . . . . 10  |-  ( ( A F B )  e.  RR  <->  A. a
( a  =  ( A F B )  ->  a  e.  RR ) )
1917, 18syl6ibr 227 . . . . . . . . 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 213 . . . . . . . 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 213 . . . . . . 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 2795 . . . . . 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 2764 . . . . . . . 8  |-  F/ k A. k  e.  ( 1 ... N ) ( A F B )  e.  RR
24 rsp 2774 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  (
k  e.  ( 1 ... N )  -> 
( A F B )  e.  RR ) )
25 eleq1a 2510 . . . . . . . . 9  |-  ( ( A F B )  e.  RR  ->  (
a  =  ( A F B )  -> 
a  e.  RR ) )
2624, 25syl6 33 . . . . . . . 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 2836 . . . . . . 7  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  ( E. k  e.  (
1 ... N ) a  =  ( A F B )  ->  a  e.  RR ) )
2827alrimiv 1690 . . . . . 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 188 . . . . 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 249 . . . 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 249 . . 3  |-  ( ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR  <->  A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
326, 31bitri 249 . 2  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
331, 32syl6bb 261 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 184   A.wal 1362    = wceq 1364    e. wcel 1761   {cab 2427   A.wral 2713   E.wrex 2714    C_ wss 3325    e. cmpt 4347   ran crn 4837    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   RRcr 9277   1c1 9279   NNcn 10318   ...cfz 11433   EEcee 23069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-ee 23072
This theorem is referenced by:  eleesub  23092  eleesubd  23093  axsegconlem1  23098  axsegconlem8  23105  axpasch  23122  axeuclidlem  23143  axcontlem2  23146
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