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Theorem cpnfval 19687
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
cpnfval  |-  ( S 
C_  CC  ->  ( C ^n `  S )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
Distinct variable group:    f, n, S

Proof of Theorem cpnfval
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 cnex 9006 . . 3  |-  CC  e.  _V
21elpw2 4307 . 2  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3 oveq2 6030 . . . . 5  |-  ( s  =  S  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
4 oveq1 6029 . . . . . . 7  |-  ( s  =  S  ->  (
s  D n f )  =  ( S  D n f ) )
54fveq1d 5672 . . . . . 6  |-  ( s  =  S  ->  (
( s  D n
f ) `  n
)  =  ( ( S  D n f ) `  n ) )
65eleq1d 2455 . . . . 5  |-  ( s  =  S  ->  (
( ( s  D n f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  D n f ) `  n )  e.  ( dom  f -cn-> CC ) ) )
73, 6rabeqbidv 2896 . . . 4  |-  ( s  =  S  ->  { 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 ) } )
87mpteq2dv 4239 . . 3  |-  ( s  =  S  ->  (
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 ) } ) )
9 df-cpn 19625 . . 3  |-  C ^n  =  ( s  e. 
~P CC  |->  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  s )  |  ( ( s  D n
f ) `  n
)  e.  ( dom  f -cn-> CC ) } ) )
10 nn0ex 10161 . . . 4  |-  NN0  e.  _V
1110mptex 5907 . . 3  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } )  e.  _V
128, 9, 11fvmpt 5747 . 2  |-  ( S  e.  ~P CC  ->  ( C ^n `  S
)  =  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
132, 12sylbir 205 1  |-  ( S 
C_  CC  ->  ( C ^n `  S )  =  ( n  e. 
NN0  |->  { 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    = wceq 1649    e. wcel 1717   {crab 2655    C_ wss 3265   ~Pcpw 3744    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:  fncpn  19688  elcpn  19689
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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