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Theorem elcpn 19812
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 19810 . . . . 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 5722 . . . 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 5720 . . . . . . 7  |-  ( n  =  N  ->  (
( S  D n
f ) `  n
)  =  ( ( S  D n f ) `  N ) )
43eleq1d 2501 . . . . . 6  |-  ( n  =  N  ->  (
( ( S  D n f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n f ) `  N )  e.  ( dom  f -cn-> CC ) ) )
54rabbidv 2940 . . . . 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 2435 . . . . 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 6098 . . . . . 6  |-  ( CC 
^pm  S )  e. 
_V
87rabex 4346 . . . . 5  |-  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  N )  e.  ( dom  f -cn->
CC ) }  e.  _V
95, 6, 8fvmpt 5798 . . . 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 2487 . . 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 2502 . 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 6081 . . . . 5  |-  ( f  =  F  ->  ( S  D n f )  =  ( S  D n F ) )
1312fveq1d 5722 . . . 4  |-  ( f  =  F  ->  (
( S  D n
f ) `  N
)  =  ( ( S  D n F ) `  N ) )
14 dmeq 5062 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
1514oveq1d 6088 . . . 4  |-  ( f  =  F  ->  ( dom  f -cn-> CC )  =  ( dom  F -cn-> CC ) )
1613, 15eleq12d 2503 . . 3  |-  ( f  =  F  ->  (
( ( S  D n f ) `  N )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) )
1716elrab 3084 . 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 1652    e. wcel 1725   {crab 2701    C_ wss 3312    e. cmpt 4258   dom cdm 4870   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   CCcc 8980   NN0cn0 10213   -cn->ccncf 18898    D ncdvn 19743   C ^nccpn 19744
This theorem is referenced by:  cpnord  19813  cpncn  19814  cpnres  19815  c1lip2  19874  plycpn  20198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-recs 6625  df-rdg 6660  df-nn 9993  df-n0 10214  df-cpn 19748
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