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Theorem dvnp1 22058
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 993 . . . . 5  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 nn0uz 11107 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2560 . . . 4  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= `  0 )
)
4 seqp1 12080 . . . 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 5869 . . . 4  |-  (  seq 0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) `
 N )  e. 
_V
7 fvex 5869 . . . 4  |-  ( ( NN0  X.  { F } ) `  ( N  +  1 ) )  e.  _V
86, 7algrflem 6884 . . 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 2519 . 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 2462 . . . . 5  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
1110dvnfval 22055 . . . 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 1011 . . 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 5861 . 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 5869 . . . 4  |-  ( ( S  Dn F ) `  N )  e.  _V
15 oveq2 6285 . . . . 5  |-  ( x  =  ( ( S  Dn F ) `
 N )  -> 
( S  _D  x
)  =  ( S  _D  ( ( S  Dn F ) `
 N ) ) )
16 ovex 6302 . . . . 5  |-  ( S  _D  ( ( S  Dn F ) `
 N ) )  e.  _V
1715, 10, 16fvmpt 5943 . . . 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 5861 . . . 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 5863 . . 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 2517 . 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 2513 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3108    C_ wss 3471   {csn 4022    |-> cmpt 4500    X. cxp 4992    o. ccom 4998   ` cfv 5581  (class class class)co 6277   1stc1st 6774    ^pm cpm 7413   CCcc 9481   0cc0 9483   1c1 9484    + caddc 9486   NN0cn0 10786   ZZ>=cuz 11073    seqcseq 12065    _D cdv 21997    Dncdvn 21998
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-seq 12066  df-dvn 22002
This theorem is referenced by:  dvn1  22059  dvnadd  22062  dvnres  22064  cpnord  22068  dvnfre  22085  c1lip2  22129  dvnply2  22412  dvntaylp  22495  taylthlem1  22497  taylthlem2  22498
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