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Theorem cpnfval 22461
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 9590 . . 3  |-  CC  e.  _V
21elpw2 4620 . 2  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3 oveq2 6304 . . . . 5  |-  ( s  =  S  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
4 oveq1 6303 . . . . . . 7  |-  ( s  =  S  ->  (
s  Dn f )  =  ( S  Dn f ) )
54fveq1d 5874 . . . . . 6  |-  ( s  =  S  ->  (
( s  Dn
f ) `  n
)  =  ( ( S  Dn f ) `  n ) )
65eleq1d 2526 . . . . 5  |-  ( s  =  S  ->  (
( ( s  Dn f ) `  n )  e.  ( dom  f -cn-> CC )  <-> 
( ( S  Dn f ) `  n )  e.  ( dom  f -cn-> CC ) ) )
73, 6rabeqbidv 3104 . . . 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 4544 . . 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 22399 . . 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 10822 . . . 4  |-  NN0  e.  _V
1110mptex 6144 . . 3  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )  e.  _V
128, 9, 11fvmpt 5956 . 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 1395    e. wcel 1819   {crab 2811    C_ wss 3471   ~Pcpw 4015    |-> cmpt 4515   dom cdm 5008   ` cfv 5594  (class class class)co 6296    ^pm cpm 7439   CCcc 9507   NN0cn0 10816   -cn->ccncf 21506    Dncdvn 22394   C^nccpn 22395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-om 6700  df-recs 7060  df-rdg 7094  df-nn 10557  df-n0 10817  df-cpn 22399
This theorem is referenced by:  fncpn  22462  elcpn  22463
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