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Theorem cpnfval 21542
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 9477 . . 3  |-  CC  e.  _V
21elpw2 4567 . 2  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3 oveq2 6211 . . . . 5  |-  ( s  =  S  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
4 oveq1 6210 . . . . . . 7  |-  ( s  =  S  ->  (
s  Dn f )  =  ( S  Dn f ) )
54fveq1d 5804 . . . . . 6  |-  ( s  =  S  ->  (
( s  Dn
f ) `  n
)  =  ( ( S  Dn f ) `  n ) )
65eleq1d 2523 . . . . 5  |-  ( s  =  S  ->  (
( ( s  Dn f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  Dn f ) `  n )  e.  ( dom  f -cn-> CC ) ) )
73, 6rabeqbidv 3073 . . . 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 4490 . . 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 21480 . . 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 10699 . . . 4  |-  NN0  e.  _V
1110mptex 6060 . . 3  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )  e.  _V
128, 9, 11fvmpt 5886 . 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 1370    e. wcel 1758   {crab 2803    C_ wss 3439   ~Pcpw 3971    |-> cmpt 4461   dom cdm 4951   ` cfv 5529  (class class class)co 6203    ^pm cpm 7328   CCcc 9394   NN0cn0 10693   -cn->ccncf 20587    Dncdvn 21475   C^nccpn 21476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-i2m1 9464  ax-1ne0 9465  ax-rrecex 9468  ax-cnre 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-om 6590  df-recs 6945  df-rdg 6979  df-nn 10437  df-n0 10694  df-cpn 21480
This theorem is referenced by:  fncpn  21543  elcpn  21544
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