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Theorem dvnfval 19271
Description: Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
dvnfval.1  |-  G  =  ( x  e.  _V  |->  ( S  _D  x
) )
Assertion
Ref Expression
dvnfval  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  D n F )  =  seq  0 ( ( G  o.  1st ) ,  ( NN0  X. 
{ F } ) ) )
Distinct variable groups:    x, F    x, S
Allowed substitution hint:    G( x)

Proof of Theorem dvnfval
Dummy variables  f 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvn 19218 . . 3  |-  D n  =  ( s  e. 
~P CC ,  f  e.  ( CC  ^pm  s )  |->  seq  0
( ( ( x  e.  _V  |->  ( s  _D  x ) )  o.  1st ) ,  ( NN0  X.  {
f } ) ) )
21a1i 10 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  D n  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s )  |->  seq  0
( ( ( x  e.  _V  |->  ( s  _D  x ) )  o.  1st ) ,  ( NN0  X.  {
f } ) ) ) )
3 simprl 732 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
s  =  S )
43oveq1d 5873 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( s  _D  x
)  =  ( S  _D  x ) )
54mpteq2dv 4107 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( x  e.  _V  |->  ( s  _D  x
) )  =  ( x  e.  _V  |->  ( S  _D  x ) ) )
6 dvnfval.1 . . . . . 6  |-  G  =  ( x  e.  _V  |->  ( S  _D  x
) )
75, 6syl6eqr 2333 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( x  e.  _V  |->  ( s  _D  x
) )  =  G )
87coeq1d 4845 . . . 4  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( ( x  e. 
_V  |->  ( s  _D  x ) )  o. 
1st )  =  ( G  o.  1st )
)
98seqeq2d 11053 . . 3  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  ->  seq  0 ( ( ( x  e.  _V  |->  ( s  _D  x ) )  o.  1st ) ,  ( NN0  X.  { f } ) )  =  seq  0
( ( G  o.  1st ) ,  ( NN0 
X.  { f } ) ) )
10 simprr 733 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
f  =  F )
1110sneqd 3653 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  ->  { f }  =  { F } )
1211xpeq2d 4713 . . . 4  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  X.  { f } )  =  ( NN0  X.  { F } ) )
1312seqeq3d 11054 . . 3  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  ->  seq  0 ( ( G  o.  1st ) ,  ( NN0  X.  {
f } ) )  =  seq  0 ( ( G  o.  1st ) ,  ( NN0  X. 
{ F } ) ) )
149, 13eqtrd 2315 . 2  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  ->  seq  0 ( ( ( x  e.  _V  |->  ( s  _D  x ) )  o.  1st ) ,  ( NN0  X.  { f } ) )  =  seq  0
( ( G  o.  1st ) ,  ( NN0 
X.  { F }
) ) )
15 simpr 447 . . 3  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  s  =  S )  ->  s  =  S )
1615oveq2d 5874 . 2  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  s  =  S )  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S )
)
17 simpl 443 . . 3  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  S  C_  CC )
18 cnex 8818 . . . 4  |-  CC  e.  _V
1918elpw2 4175 . . 3  |-  ( S  e.  ~P CC  <->  S  C_  CC )
2017, 19sylibr 203 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  S  e.  ~P CC )
21 simpr 447 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  F  e.  ( CC  ^pm  S
) )
22 seqex 11048 . . 3  |-  seq  0
( ( G  o.  1st ) ,  ( NN0 
X.  { F }
) )  e.  _V
2322a1i 10 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  seq  0 ( ( G  o.  1st ) ,  ( NN0  X.  { F } ) )  e. 
_V )
242, 14, 16, 20, 21, 23ovmpt2dx 5974 1  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  D n F )  =  seq  0 ( ( G  o.  1st ) ,  ( NN0  X. 
{ F } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   {csn 3640    e. cmpt 4077    X. cxp 4687    o. ccom 4693  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120    ^pm cpm 6773   CCcc 8735   0cc0 8737   NN0cn0 9965    seq cseq 11046    _D cdv 19213    D ncdvn 19214
This theorem is referenced by:  dvnff  19272  dvn0  19273  dvnp1  19274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-dvn 19218
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