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Theorem dvnp1 22622
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 1001 . . . . 5  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 nn0uz 11163 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2502 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= `  0 )
)
4 seqp1 12168 . . . 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 17 . . 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 5861 . . . 4  |-  (  seq 0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N )  e. 
_V
7 fvex 5861 . . . 4  |-  ( ( NN0  X.  { F } ) `  ( N  +  1 ) )  e.  _V
86, 7algrflem 6895 . . 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 2461 . 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 2404 . . . . 5  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
1110dvnfval 22619 . . . 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 1019 . . 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 5853 . 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 5861 . . . 4  |-  ( ( S  Dn F ) `  N )  e.  _V
15 oveq2 6288 . . . . 5  |-  ( x  =  ( ( S  Dn F ) `
 N )  -> 
( S  _D  x
)  =  ( S  _D  ( ( S  Dn F ) `
 N ) ) )
16 ovex 6308 . . . . 5  |-  ( S  _D  ( ( S  Dn F ) `
 N ) )  e.  _V
1715, 10, 16fvmpt 5934 . . . 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 5853 . . . 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 5855 . . 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 2459 . 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 2455 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 976    = wceq 1407    e. wcel 1844   _Vcvv 3061    C_ wss 3416   {csn 3974    |-> cmpt 4455    X. cxp 4823    o. ccom 4829   ` cfv 5571  (class class class)co 6280   1stc1st 6784    ^pm cpm 7460   CCcc 9522   0cc0 9524   1c1 9525    + caddc 9527   NN0cn0 10838   ZZ>=cuz 11129    seqcseq 12153    _D cdv 22561    Dncdvn 22562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-seq 12154  df-dvn 22566
This theorem is referenced by:  dvn1  22623  dvnadd  22626  dvnres  22628  cpnord  22632  dvnfre  22649  c1lip2  22693  dvnply2  22977  dvntaylp  23060  taylthlem1  23062  taylthlem2  23063  dvnmptdivc  37116  dvnmptconst  37119  dvnxpaek  37120  dvnmul  37121  etransclem2  37400
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