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Theorem dvnfval 19800
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 19747 . . 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 6088 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( s  _D  x
)  =  ( S  _D  x ) )
54mpteq2dv 4288 . . . . . 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 2485 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( x  e.  _V  |->  ( s  _D  x
) )  =  G )
87coeq1d 5026 . . . 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 11322 . . 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 3819 . . . . 5  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  ->  { f }  =  { F } )
1211xpeq2d 4894 . . . 4  |-  ( ( ( S  C_  CC  /\  F  e.  ( CC 
^pm  S ) )  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  X.  { f } )  =  ( NN0  X.  { F } ) )
1312seqeq3d 11323 . . 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 2467 . 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 6089 . 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 9063 . . . 4  |-  CC  e.  _V
1918elpw2 4356 . . 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 11317 . . 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 6192 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 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   {csn 3806    e. cmpt 4258    X. cxp 4868    o. ccom 4874  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339    ^pm cpm 7011   CCcc 8980   0cc0 8982   NN0cn0 10213    seq cseq 11315    _D cdv 19742    D ncdvn 19743
This theorem is referenced by:  dvnff  19801  dvn0  19802  dvnp1  19803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-seq 11316  df-dvn 19747
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