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Theorem cpnfval 21381
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  Dn 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 9355 . . 3  |-  CC  e.  _V
21elpw2 4451 . 2  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3 oveq2 6094 . . . . 5  |-  ( s  =  S  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
4 oveq1 6093 . . . . . . 7  |-  ( s  =  S  ->  (
s  Dn f )  =  ( S  Dn f ) )
54fveq1d 5688 . . . . . 6  |-  ( s  =  S  ->  (
( s  Dn
f ) `  n
)  =  ( ( S  Dn f ) `  n ) )
65eleq1d 2504 . . . . 5  |-  ( s  =  S  ->  (
( ( s  Dn f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  Dn f ) `  n )  e.  ( dom  f -cn-> CC ) ) )
73, 6rabeqbidv 2962 . . . 4  |-  ( s  =  S  ->  { f  e.  ( CC  ^pm  s )  |  ( ( s  Dn
f ) `  n
)  e.  ( dom  f -cn-> CC ) }  =  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `
 n )  e.  ( dom  f -cn-> CC ) } )
87mpteq2dv 4374 . . 3  |-  ( s  =  S  ->  (
n  e.  NN0  |->  { f  e.  ( CC  ^pm  s )  |  ( ( s  Dn
f ) `  n
)  e.  ( dom  f -cn-> CC ) } )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
9 df-cpn 21319 . . 3  |-  C^n  =  ( s  e. 
~P CC  |->  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  s )  |  ( ( s  Dn
f ) `  n
)  e.  ( dom  f -cn-> CC ) } ) )
10 nn0ex 10577 . . . 4  |-  NN0  e.  _V
1110mptex 5943 . . 3  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )  e.  _V
128, 9, 11fvmpt 5769 . 2  |-  ( S  e.  ~P CC  ->  ( C^n `  S
)  =  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
132, 12sylbir 213 1  |-  ( S 
C_  CC  ->  ( C^n `  S )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2714    C_ wss 3323   ~Pcpw 3855    e. cmpt 4345   dom cdm 4835   ` cfv 5413  (class class class)co 6086    ^pm cpm 7207   CCcc 9272   NN0cn0 10571   -cn->ccncf 20427    Dncdvn 21314   C^nccpn 21315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-i2m1 9342  ax-1ne0 9343  ax-rrecex 9346  ax-cnre 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-om 6472  df-recs 6824  df-rdg 6858  df-nn 10315  df-n0 10572  df-cpn 21319
This theorem is referenced by:  fncpn  21382  elcpn  21383
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