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Theorem elcpn 19299
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 19297 . . . . 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 5543 . . . 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 5541 . . . . . . 7  |-  ( n  =  N  ->  (
( S  D n
f ) `  n
)  =  ( ( S  D n f ) `  N ) )
43eleq1d 2362 . . . . . 6  |-  ( n  =  N  ->  (
( ( S  D n f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n f ) `  N )  e.  ( dom  f -cn-> CC ) ) )
54rabbidv 2793 . . . . 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 2296 . . . . 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 5899 . . . . . 6  |-  ( CC 
^pm  S )  e. 
_V
87rabex 4181 . . . . 5  |-  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  N )  e.  ( dom  f -cn->
CC ) }  e.  _V
95, 6, 8fvmpt 5618 . . . 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 2348 . . 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 2363 . 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 5882 . . . . 5  |-  ( f  =  F  ->  ( S  D n f )  =  ( S  D n F ) )
1312fveq1d 5543 . . . 4  |-  ( f  =  F  ->  (
( S  D n
f ) `  N
)  =  ( ( S  D n F ) `  N ) )
14 dmeq 4895 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
1514oveq1d 5889 . . . 4  |-  ( f  =  F  ->  ( dom  f -cn-> CC )  =  ( dom  F -cn-> CC ) )
1613, 15eleq12d 2364 . . 3  |-  ( f  =  F  ->  (
( ( S  D n f ) `  N )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) )
1716elrab 2936 . 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 1632    e. wcel 1696   {crab 2560    C_ wss 3165    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   NN0cn0 9981   -cn->ccncf 18396    D ncdvn 19230   C ^nccpn 19231
This theorem is referenced by:  cpnord  19300  cpncn  19301  cpnres  19302  c1lip2  19361  plycpn  19685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-nn 9763  df-n0 9982  df-cpn 19235
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