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Theorem elcpn 19283
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  D n 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 19281 . . . . 5  |-  ( S 
C_  CC  ->  ( C ^n `  S )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
21fveq1d 5527 . . . 4  |-  ( S 
C_  CC  ->  ( ( C ^n `  S
) `  N )  =  ( ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } ) `
 N ) )
3 fveq2 5525 . . . . . . 7  |-  ( n  =  N  ->  (
( S  D n
f ) `  n
)  =  ( ( S  D n f ) `  N ) )
43eleq1d 2349 . . . . . 6  |-  ( n  =  N  ->  (
( ( S  D n f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n f ) `  N )  e.  ( dom  f -cn-> CC ) ) )
54rabbidv 2780 . . . . 5  |-  ( n  =  N  ->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) }  =  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `
 N )  e.  ( dom  f -cn-> CC ) } )
6 eqid 2283 . . . . 5  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } )
7 ovex 5883 . . . . . 6  |-  ( CC 
^pm  S )  e. 
_V
87rabex 4165 . . . . 5  |-  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  N )  e.  ( dom  f -cn->
CC ) }  e.  _V
95, 6, 8fvmpt 5602 . . . 4  |-  ( N  e.  NN0  ->  ( ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } ) `
 N )  =  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `
 N )  e.  ( dom  f -cn-> CC ) } )
102, 9sylan9eq 2335 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  (
( C ^n `  S ) `  N
)  =  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  N )  e.  ( dom  f -cn->
CC ) } )
1110eleq2d 2350 . 2  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  ( F  e.  ( (
C ^n `  S
) `  N )  <->  F  e.  { f  e.  ( CC  ^pm  S
)  |  ( ( S  D n f ) `  N )  e.  ( dom  f -cn->
CC ) } ) )
12 oveq2 5866 . . . . 5  |-  ( f  =  F  ->  ( S  D n f )  =  ( S  D n F ) )
1312fveq1d 5527 . . . 4  |-  ( f  =  F  ->  (
( S  D n
f ) `  N
)  =  ( ( S  D n F ) `  N ) )
14 dmeq 4879 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
1514oveq1d 5873 . . . 4  |-  ( f  =  F  ->  ( dom  f -cn-> CC )  =  ( dom  F -cn-> CC ) )
1613, 15eleq12d 2351 . . 3  |-  ( f  =  F  ->  (
( ( S  D n f ) `  N )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) )
1716elrab 2923 . 2  |-  ( F  e.  { f  e.  ( CC  ^pm  S
)  |  ( ( S  D n f ) `  N )  e.  ( dom  f -cn->
CC ) }  <->  ( F  e.  ( CC  ^pm  S
)  /\  ( ( S  D n F ) `
 N )  e.  ( dom  F -cn-> CC ) ) )
1811, 17syl6bb 252 1  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  ( F  e.  ( (
C ^n `  S
) `  N )  <->  ( F  e.  ( CC 
^pm  S )  /\  ( ( S  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   NN0cn0 9965   -cn->ccncf 18380    D ncdvn 19214   C ^nccpn 19215
This theorem is referenced by:  cpnord  19284  cpncn  19285  cpnres  19286  c1lip2  19345  plycpn  19669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-recs 6388  df-rdg 6423  df-nn 9747  df-n0 9966  df-cpn 19219
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