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Theorem dvnfval 19813
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 19760 . . 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 734 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
s  =  S )
43oveq1d 6099 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( s  _D  x
)  =  ( S  _D  x ) )
54mpteq2dv 4299 . . . . . 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 2488 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( x  e.  _V  |->  ( s  _D  x
) )  =  G )
87coeq1d 5037 . . . 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 11335 . . 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 735 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
f  =  F )
1110sneqd 3829 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  ->  { f }  =  { F } )
1211xpeq2d 4905 . . . 4  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  X.  { f } )  =  ( NN0  X.  { F } ) )
1312seqeq3d 11336 . . 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 2470 . 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 449 . . 3  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  s  =  S )  ->  s  =  S )
1615oveq2d 6100 . 2  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  s  =  S )  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S )
)
17 simpl 445 . . 3  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  S  C_  CC )
18 cnex 9076 . . . 4  |-  CC  e.  _V
1918elpw2 4367 . . 3  |-  ( S  e.  ~P CC  <->  S  C_  CC )
2017, 19sylibr 205 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  S  e.  ~P CC )
21 simpr 449 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  F  e.  ( CC  ^pm  S
) )
22 seqex 11330 . . 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 6203 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   {csn 3816    e. cmpt 4269    X. cxp 4879    o. ccom 4885  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350    ^pm cpm 7022   CCcc 8993   0cc0 8995   NN0cn0 10226    seq cseq 11328    _D cdv 19755    D ncdvn 19756
This theorem is referenced by:  dvnff  19814  dvn0  19815  dvnp1  19816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-recs 6636  df-rdg 6671  df-seq 11329  df-dvn 19760
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