MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptelee Structured version   Unicode version

Theorem mptelee 23146
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 23145 . 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 6121 . . . . 5  |-  ( A F B )  e. 
_V
3 eqid 2443 . . . . 5  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  =  ( k  e.  ( 1 ... N
)  |->  ( A F B ) )
42, 3fnmpti 5544 . . . 4  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  Fn  ( 1 ... N )
5 df-f 5427 . . . 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 909 . . 3  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR )
73rnmpt 5090 . . . . 5  |-  ran  (
k  e.  ( 1 ... N )  |->  ( A F B ) )  =  { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }
87sseq1i 3385 . . . 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 3426 . . . . 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 2777 . . . . . . . . 9  |-  F/ k E. k  e.  ( 1 ... N ) a  =  ( A F B )
11 nfv 1673 . . . . . . . . 9  |-  F/ k  a  e.  RR
1210, 11nfim 1853 . . . . . . . 8  |-  F/ k ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )
1312nfal 1873 . . . . . . 7  |-  F/ k A. a ( E. k  e.  ( 1 ... N ) a  =  ( A F B )  ->  a  e.  RR )
14 r19.23v 2838 . . . . . . . . 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 1610 . . . . . . . 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 2996 . . . . . . . . 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 2781 . . . . . . . . . 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 3101 . . . . . . . . . 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 2802 . . . . . 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 2771 . . . . . . . 8  |-  F/ k A. k  e.  ( 1 ... N ) ( A F B )  e.  RR
24 rsp 2781 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  (
k  e.  ( 1 ... N )  -> 
( A F B )  e.  RR ) )
25 eleq1a 2512 . . . . . . . . 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 2843 . . . . . . 7  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  ( E. k  e.  (
1 ... N ) a  =  ( A F B )  ->  a  e.  RR ) )
2827alrimiv 1685 . . . . . 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 1367    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721    C_ wss 3333    e. cmpt 4355   ran crn 4846    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   RRcr 9286   1c1 9288   NNcn 10327   ...cfz 11442   EEcee 23139
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-ee 23142
This theorem is referenced by:  eleesub  23162  eleesubd  23163  axsegconlem1  23168  axsegconlem8  23175  axpasch  23192  axeuclidlem  23213  axcontlem2  23216
  Copyright terms: Public domain W3C validator