MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvnp1 Structured version   Unicode version

Theorem dvnp1 19842
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 960 . . . . 5  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 nn0uz 10551 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2532 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= `  0 )
)
4 seqp1 11369 . . . 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 5771 . . . 4  |-  (  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N )  e. 
_V
7 fvex 5771 . . . 4  |-  ( ( NN0  X.  { F } ) `  ( N  +  1 ) )  e.  _V
86, 7algrflem 6484 . . 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 2490 . 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 2442 . . . . 5  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
1110dvnfval 19839 . . . 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 978 . . 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 5759 . 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 5771 . . . 4  |-  ( ( S  D n F ) `  N )  e.  _V
15 oveq2 6118 . . . . 5  |-  ( x  =  ( ( S  D n F ) `
 N )  -> 
( S  _D  x
)  =  ( S  _D  ( ( S  D n F ) `
 N ) ) )
16 ovex 6135 . . . . 5  |-  ( S  _D  ( ( S  D n F ) `
 N ) )  e.  _V
1715, 10, 16fvmpt 5835 . . . 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 5 . . 3  |-  ( ( x  e.  _V  |->  ( S  _D  x ) ) `  ( ( S  D n F ) `  N ) )  =  ( S  _D  ( ( S  D n F ) `
 N ) )
1912fveq1d 5759 . . . 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 5761 . . 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 2488 . 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 2484 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 937    = wceq 1653    e. wcel 1727   _Vcvv 2962    C_ wss 3306   {csn 3838    e. cmpt 4291    X. cxp 4905    o. ccom 4911   ` cfv 5483  (class class class)co 6110   1stc1st 6376    ^pm cpm 7048   CCcc 9019   0cc0 9021   1c1 9022    + caddc 9024   NN0cn0 10252   ZZ>=cuz 10519    seq cseq 11354    _D cdv 19781    D ncdvn 19782
This theorem is referenced by:  dvn1  19843  dvnadd  19846  dvnres  19848  cpnord  19852  dvnfre  19869  c1lip2  19913  dvnply2  20235  dvntaylp  20318  taylthlem1  20320  taylthlem2  20321
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 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-seq 11355  df-dvn 19786
  Copyright terms: Public domain W3C validator