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Theorem dvnfval 19287
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 19234 . . 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 5889 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( s  _D  x
)  =  ( S  _D  x ) )
54mpteq2dv 4123 . . . . . 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 2346 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( x  e.  _V  |->  ( s  _D  x
) )  =  G )
87coeq1d 4861 . . . 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 11069 . . 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 3666 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  ->  { f }  =  { F } )
1211xpeq2d 4729 . . . 4  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  X.  { f } )  =  ( NN0  X.  { F } ) )
1312seqeq3d 11070 . . 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 2328 . 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 5890 . 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 8834 . . . 4  |-  CC  e.  _V
1918elpw2 4191 . . 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 11064 . . 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 5990 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   {csn 3653    e. cmpt 4093    X. cxp 4703    o. ccom 4709  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136    ^pm cpm 6789   CCcc 8751   0cc0 8753   NN0cn0 9981    seq cseq 11062    _D cdv 19229    D ncdvn 19230
This theorem is referenced by:  dvnff  19288  dvn0  19289  dvnp1  19290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-dvn 19234
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