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Theorem dvnfval 19668
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 19615 . . 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 11 . 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 733 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
s  =  S )
43oveq1d 6028 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( s  _D  x
)  =  ( S  _D  x ) )
54mpteq2dv 4230 . . . . . 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 2430 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( x  e.  _V  |->  ( s  _D  x
) )  =  G )
87coeq1d 4967 . . . 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 11250 . . 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 734 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
f  =  F )
1110sneqd 3763 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  ->  { f }  =  { F } )
1211xpeq2d 4835 . . . 4  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  X.  { f } )  =  ( NN0  X.  { F } ) )
1312seqeq3d 11251 . . 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 2412 . 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 448 . . 3  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  s  =  S )  ->  s  =  S )
1615oveq2d 6029 . 2  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  s  =  S )  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S )
)
17 simpl 444 . . 3  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  S  C_  CC )
18 cnex 8997 . . . 4  |-  CC  e.  _V
1918elpw2 4298 . . 3  |-  ( S  e.  ~P CC  <->  S  C_  CC )
2017, 19sylibr 204 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  S  e.  ~P CC )
21 simpr 448 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  F  e.  ( CC  ^pm  S
) )
22 seqex 11245 . . 3  |-  seq  0
( ( G  o.  1st ) ,  ( NN0 
X.  { F }
) )  e.  _V
2322a1i 11 . 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 6132 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 359    = wceq 1649    e. wcel 1717   _Vcvv 2892    C_ wss 3256   ~Pcpw 3735   {csn 3750    e. cmpt 4200    X. cxp 4809    o. ccom 4815  (class class class)co 6013    e. cmpt2 6015   1stc1st 6279    ^pm cpm 6948   CCcc 8914   0cc0 8916   NN0cn0 10146    seq cseq 11243    _D cdv 19610    D ncdvn 19611
This theorem is referenced by:  dvnff  19669  dvn0  19670  dvnp1  19671
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-recs 6562  df-rdg 6597  df-seq 11244  df-dvn 19615
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