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Theorem dvnp1 19274
Description: Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
dvnp1  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( ( S  D n F ) `
 ( N  + 
1 ) )  =  ( S  _D  (
( S  D n F ) `  N
) ) )

Proof of Theorem dvnp1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3 957 . . . . 5  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 nn0uz 10262 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2373 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= `  0 )
)
4 seqp1 11061 . . . 4  |-  ( N  e.  ( ZZ>= `  0
)  ->  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 ( N  + 
1 ) )  =  ( (  seq  0
( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `  N ) ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st )
( ( NN0  X.  { F } ) `  ( N  +  1
) ) ) )
53, 4syl 15 . . 3  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 ( N  + 
1 ) )  =  ( (  seq  0
( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `  N ) ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st )
( ( NN0  X.  { F } ) `  ( N  +  1
) ) ) )
6 fvex 5539 . . . 4  |-  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N )  e. 
_V
7 fvex 5539 . . . 4  |-  ( ( NN0  X.  { F } ) `  ( N  +  1 ) )  e.  _V
86, 7algrflem 6224 . . 3  |-  ( (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x
) )  o.  1st ) ,  ( NN0  X. 
{ F } ) ) `  N ) ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ( ( NN0  X.  { F } ) `  ( N  +  1 ) ) )  =  ( ( x  e.  _V  |->  ( S  _D  x
) ) `  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N ) )
95, 8syl6eq 2331 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 ( N  + 
1 ) )  =  ( ( x  e. 
_V  |->  ( S  _D  x ) ) `  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x
) )  o.  1st ) ,  ( NN0  X. 
{ F } ) ) `  N ) ) )
10 eqid 2283 . . . . 5  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
1110dvnfval 19271 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  D n F )  =  seq  0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) )
12113adant3 975 . . 3  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( S  D n F )  =  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x
) )  o.  1st ) ,  ( NN0  X. 
{ F } ) ) )
1312fveq1d 5527 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( ( S  D n F ) `
 ( N  + 
1 ) )  =  (  seq  0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) `  ( N  +  1
) ) )
14 fvex 5539 . . . 4  |-  ( ( S  D n F ) `  N )  e.  _V
15 oveq2 5866 . . . . 5  |-  ( x  =  ( ( S  D n F ) `
 N )  -> 
( S  _D  x
)  =  ( S  _D  ( ( S  D n F ) `
 N ) ) )
16 ovex 5883 . . . . 5  |-  ( S  _D  ( ( S  D n F ) `
 N ) )  e.  _V
1715, 10, 16fvmpt 5602 . . . 4  |-  ( ( ( S  D n F ) `  N
)  e.  _V  ->  ( ( x  e.  _V  |->  ( S  _D  x
) ) `  (
( S  D n F ) `  N
) )  =  ( S  _D  ( ( S  D n F ) `  N ) ) )
1814, 17ax-mp 8 . . 3  |-  ( ( x  e.  _V  |->  ( S  _D  x ) ) `  ( ( S  D n F ) `  N ) )  =  ( S  _D  ( ( S  D n F ) `
 N ) )
1912fveq1d 5527 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( ( S  D n F ) `
 N )  =  (  seq  0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) `  N ) )
2019fveq2d 5529 . . 3  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( (
x  e.  _V  |->  ( S  _D  x ) ) `  ( ( S  D n F ) `  N ) )  =  ( ( x  e.  _V  |->  ( S  _D  x ) ) `  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N ) ) )
2118, 20syl5eqr 2329 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( S  _D  ( ( S  D n F ) `  N
) )  =  ( ( x  e.  _V  |->  ( S  _D  x
) ) `  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N ) ) )
229, 13, 213eqtr4d 2325 1  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( ( S  D n F ) `
 ( N  + 
1 ) )  =  ( S  _D  (
( S  D n F ) `  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640    e. cmpt 4077    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858   1stc1st 6120    ^pm cpm 6773   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740   NN0cn0 9965   ZZ>=cuz 10230    seq cseq 11046    _D cdv 19213    D ncdvn 19214
This theorem is referenced by:  dvn1  19275  dvnadd  19278  dvnres  19280  cpnord  19284  dvnfre  19301  c1lip2  19345  dvnply2  19667  dvntaylp  19750  taylthlem1  19752  taylthlem2  19753
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  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-nel 2449  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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-dvn 19218
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