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Theorem dvnp1 21358
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  Dn F ) `
 ( N  + 
1 ) )  =  ( S  _D  (
( S  Dn
F ) `  N
) ) )

Proof of Theorem dvnp1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3 985 . . . . 5  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 nn0uz 10891 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2531 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= `  0 )
)
4 seqp1 11817 . . . 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 16 . . 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 5698 . . . 4  |-  (  seq 0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N )  e. 
_V
7 fvex 5698 . . . 4  |-  ( ( NN0  X.  { F } ) `  ( N  +  1 ) )  e.  _V
86, 7algrflem 6680 . . 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 2489 . 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 2441 . . . . 5  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
1110dvnfval 21355 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  Dn F )  =  seq 0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) )
12113adant3 1003 . . 3  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( S  Dn F )  =  seq 0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) )
1312fveq1d 5690 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( ( S  Dn F ) `
 ( N  + 
1 ) )  =  (  seq 0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) `  ( N  +  1
) ) )
14 fvex 5698 . . . 4  |-  ( ( S  Dn F ) `  N )  e.  _V
15 oveq2 6098 . . . . 5  |-  ( x  =  ( ( S  Dn F ) `
 N )  -> 
( S  _D  x
)  =  ( S  _D  ( ( S  Dn F ) `
 N ) ) )
16 ovex 6115 . . . . 5  |-  ( S  _D  ( ( S  Dn F ) `
 N ) )  e.  _V
1715, 10, 16fvmpt 5771 . . . 4  |-  ( ( ( S  Dn
F ) `  N
)  e.  _V  ->  ( ( x  e.  _V  |->  ( S  _D  x
) ) `  (
( S  Dn
F ) `  N
) )  =  ( S  _D  ( ( S  Dn F ) `  N ) ) )
1814, 17ax-mp 5 . . 3  |-  ( ( x  e.  _V  |->  ( S  _D  x ) ) `  ( ( S  Dn F ) `  N ) )  =  ( S  _D  ( ( S  Dn F ) `
 N ) )
1912fveq1d 5690 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( ( S  Dn F ) `
 N )  =  (  seq 0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) `  N ) )
2019fveq2d 5692 . . 3  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( (
x  e.  _V  |->  ( S  _D  x ) ) `  ( ( S  Dn F ) `  N ) )  =  ( ( x  e.  _V  |->  ( S  _D  x ) ) `  (  seq 0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N ) ) )
2118, 20syl5eqr 2487 . 2  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( S  _D  ( ( S  Dn 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 2483 1  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  ( ( S  Dn F ) `
 ( N  + 
1 ) )  =  ( S  _D  (
( S  Dn
F ) `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   {csn 3874    e. cmpt 4347    X. cxp 4834    o. ccom 4840   ` cfv 5415  (class class class)co 6090   1stc1st 6574    ^pm cpm 7211   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281   NN0cn0 10575   ZZ>=cuz 10857    seqcseq 11802    _D cdv 21297    Dncdvn 21298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-seq 11803  df-dvn 21302
This theorem is referenced by:  dvn1  21359  dvnadd  21362  dvnres  21364  cpnord  21368  dvnfre  21385  c1lip2  21429  dvnply2  21712  dvntaylp  21795  taylthlem1  21797  taylthlem2  21798
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