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Theorem elcpn 19689
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 19687 . . . . 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 5672 . . . 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 5670 . . . . . . 7  |-  ( n  =  N  ->  (
( S  D n
f ) `  n
)  =  ( ( S  D n f ) `  N ) )
43eleq1d 2455 . . . . . 6  |-  ( n  =  N  ->  (
( ( S  D n f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n f ) `  N )  e.  ( dom  f -cn-> CC ) ) )
54rabbidv 2893 . . . . 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 2389 . . . . 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 6047 . . . . . 6  |-  ( CC 
^pm  S )  e. 
_V
87rabex 4297 . . . . 5  |-  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  N )  e.  ( dom  f -cn->
CC ) }  e.  _V
95, 6, 8fvmpt 5747 . . . 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 2441 . . 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 2456 . 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 6030 . . . . 5  |-  ( f  =  F  ->  ( S  D n f )  =  ( S  D n F ) )
1312fveq1d 5672 . . . 4  |-  ( f  =  F  ->  (
( S  D n
f ) `  N
)  =  ( ( S  D n F ) `  N ) )
14 dmeq 5012 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
1514oveq1d 6037 . . . 4  |-  ( f  =  F  ->  ( dom  f -cn-> CC )  =  ( dom  F -cn-> CC ) )
1613, 15eleq12d 2457 . . 3  |-  ( f  =  F  ->  (
( ( S  D n f ) `  N )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) )
1716elrab 3037 . 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 253 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2655    C_ wss 3265    e. cmpt 4209   dom cdm 4820   ` cfv 5396  (class class class)co 6022    ^pm cpm 6957   CCcc 8923   NN0cn0 10155   -cn->ccncf 18779    D ncdvn 19620   C ^nccpn 19621
This theorem is referenced by:  cpnord  19690  cpncn  19691  cpnres  19692  c1lip2  19751  plycpn  20075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-i2m1 8993  ax-1ne0 8994  ax-rrecex 8997  ax-cnre 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-recs 6571  df-rdg 6606  df-nn 9935  df-n0 10156  df-cpn 19625
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