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Theorem elcpn 22629
Description: Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
elcpn  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  ( F  e.  ( (
C^n `  S
) `  N )  <->  ( F  e.  ( CC 
^pm  S )  /\  ( ( S  Dn F ) `  N )  e.  ( dom  F -cn-> CC ) ) ) )

Proof of Theorem elcpn
Dummy variables  f  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cpnfval 22627 . . . . 5  |-  ( S 
C_  CC  ->  ( C^n `  S )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
21fveq1d 5851 . . . 4  |-  ( S 
C_  CC  ->  ( ( C^n `  S
) `  N )  =  ( ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } ) `
 N ) )
3 fveq2 5849 . . . . . . 7  |-  ( n  =  N  ->  (
( S  Dn
f ) `  n
)  =  ( ( S  Dn f ) `  N ) )
43eleq1d 2471 . . . . . 6  |-  ( n  =  N  ->  (
( ( S  Dn f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  Dn f ) `  N )  e.  ( dom  f -cn-> CC ) ) )
54rabbidv 3051 . . . . 5  |-  ( n  =  N  ->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) }  =  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `
 N )  e.  ( dom  f -cn-> CC ) } )
6 eqid 2402 . . . . 5  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )
7 ovex 6306 . . . . . 6  |-  ( CC 
^pm  S )  e. 
_V
87rabex 4545 . . . . 5  |-  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  N )  e.  ( dom  f -cn->
CC ) }  e.  _V
95, 6, 8fvmpt 5932 . . . 4  |-  ( N  e.  NN0  ->  ( ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } ) `
 N )  =  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `
 N )  e.  ( dom  f -cn-> CC ) } )
102, 9sylan9eq 2463 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  (
( C^n `  S ) `  N
)  =  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  N )  e.  ( dom  f -cn->
CC ) } )
1110eleq2d 2472 . 2  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  ( F  e.  ( (
C^n `  S
) `  N )  <->  F  e.  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  N )  e.  ( dom  f -cn->
CC ) } ) )
12 oveq2 6286 . . . . 5  |-  ( f  =  F  ->  ( S  Dn f )  =  ( S  Dn F ) )
1312fveq1d 5851 . . . 4  |-  ( f  =  F  ->  (
( S  Dn
f ) `  N
)  =  ( ( S  Dn F ) `  N ) )
14 dmeq 5024 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
1514oveq1d 6293 . . . 4  |-  ( f  =  F  ->  ( dom  f -cn-> CC )  =  ( dom  F -cn-> CC ) )
1613, 15eleq12d 2484 . . 3  |-  ( f  =  F  ->  (
( ( S  Dn f ) `  N )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  Dn F ) `  N )  e.  ( dom  F -cn-> CC ) ) )
1716elrab 3207 . 2  |-  ( F  e.  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  N )  e.  ( dom  f -cn->
CC ) }  <->  ( F  e.  ( CC  ^pm  S
)  /\  ( ( S  Dn F ) `
 N )  e.  ( dom  F -cn-> CC ) ) )
1811, 17syl6bb 261 1  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  ( F  e.  ( (
C^n `  S
) `  N )  <->  ( F  e.  ( CC 
^pm  S )  /\  ( ( S  Dn F ) `  N )  e.  ( dom  F -cn-> CC ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   {crab 2758    C_ wss 3414    |-> cmpt 4453   dom cdm 4823   ` cfv 5569  (class class class)co 6278    ^pm cpm 7458   CCcc 9520   NN0cn0 10836   -cn->ccncf 21672    Dncdvn 22560   C^nccpn 22561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-i2m1 9590  ax-1ne0 9591  ax-rrecex 9594  ax-cnre 9595
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-nn 10577  df-n0 10837  df-cpn 22565
This theorem is referenced by:  cpnord  22630  cpncn  22631  cpnres  22632  c1lip2  22691  plycpn  22977
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