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Theorem dvnff 22452
Description: The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
dvnff  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  Dn F ) : NN0 --> ( CC 
^pm  dom  F ) )

Proof of Theorem dvnff
Dummy variables  k  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 11140 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 10897 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  0  e.  ZZ )
3 fvconst2g 6126 . . . . 5  |-  ( ( F  e.  ( CC 
^pm  S )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  =  F )
43adantll 713 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  =  F )
5 dmexg 6730 . . . . . 6  |-  ( F  e.  ( CC  ^pm  S )  ->  dom  F  e. 
_V )
65ad2antlr 726 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  dom  F  e.  _V )
7 cnex 9590 . . . . . 6  |-  CC  e.  _V
87a1i 11 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  CC  e.  _V )
9 elpm2g 7454 . . . . . . . . 9  |-  ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  -> 
( F  e.  ( CC  ^pm  S )  <->  ( F : dom  F --> CC  /\  dom  F  C_  S ) ) )
107, 9mpan 670 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  ( F  e.  ( CC  ^pm  S
)  <->  ( F : dom  F --> CC  /\  dom  F 
C_  S ) ) )
1110biimpa 484 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  S )
)
1211simpld 459 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  F : dom  F --> CC )
1312adantr 465 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  F : dom  F --> CC )
14 fpmg 7463 . . . . 5  |-  ( ( dom  F  e.  _V  /\  CC  e.  _V  /\  F : dom  F --> CC )  ->  F  e.  ( CC  ^pm  dom  F ) )
156, 8, 13, 14syl3anc 1228 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  F  e.  ( CC  ^pm 
dom  F ) )
164, 15eqeltrd 2545 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  e.  ( CC  ^pm  dom  F ) )
17 vex 3112 . . . . . 6  |-  k  e. 
_V
18 vex 3112 . . . . . 6  |-  n  e. 
_V
1917, 18algrflem 6908 . . . . 5  |-  ( k ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) n )  =  ( ( x  e.  _V  |->  ( S  _D  x ) ) `
 k )
20 oveq2 6304 . . . . . . 7  |-  ( x  =  k  ->  ( S  _D  x )  =  ( S  _D  k
) )
21 eqid 2457 . . . . . . 7  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
22 ovex 6324 . . . . . . 7  |-  ( S  _D  k )  e. 
_V
2320, 21, 22fvmpt 5956 . . . . . 6  |-  ( k  e.  _V  ->  (
( x  e.  _V  |->  ( S  _D  x
) ) `  k
)  =  ( S  _D  k ) )
2417, 23ax-mp 5 . . . . 5  |-  ( ( x  e.  _V  |->  ( S  _D  x ) ) `  k )  =  ( S  _D  k )
2519, 24eqtri 2486 . . . 4  |-  ( k ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) n )  =  ( S  _D  k )
267a1i 11 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  CC  e.  _V )
275ad2antlr 726 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  dom  F  e.  _V )
28 dvfg 22436 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  k ) : dom  ( S  _D  k
) --> CC )
2928ad2antrr 725 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  ( S  _D  k ) : dom  ( S  _D  k ) --> CC )
30 recnprss 22434 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
3130ad2antrr 725 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  S  C_  CC )
32 simprl 756 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  k  e.  ( CC  ^pm  dom  F ) )
33 elpm2g 7454 . . . . . . . . . 10  |-  ( ( CC  e.  _V  /\  dom  F  e.  _V )  ->  ( k  e.  ( CC  ^pm  dom  F )  <-> 
( k : dom  k
--> CC  /\  dom  k  C_ 
dom  F ) ) )
347, 27, 33sylancr 663 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  (
k  e.  ( CC 
^pm  dom  F )  <->  ( k : dom  k --> CC  /\  dom  k  C_  dom  F
) ) )
3532, 34mpbid 210 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  (
k : dom  k --> CC  /\  dom  k  C_  dom  F ) )
3635simpld 459 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  k : dom  k --> CC )
3735simprd 463 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  dom  k  C_  dom  F )
3811simprd 463 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  dom  F 
C_  S )
3938adantr 465 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  dom  F 
C_  S )
4037, 39sstrd 3509 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  dom  k  C_  S )
4131, 36, 40dvbss 22431 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  dom  ( S  _D  k
)  C_  dom  k )
4241, 37sstrd 3509 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  dom  ( S  _D  k
)  C_  dom  F )
43 elpm2r 7455 . . . . 5  |-  ( ( ( CC  e.  _V  /\ 
dom  F  e.  _V )  /\  ( ( S  _D  k ) : dom  ( S  _D  k ) --> CC  /\  dom  ( S  _D  k
)  C_  dom  F ) )  ->  ( S  _D  k )  e.  ( CC  ^pm  dom  F ) )
4426, 27, 29, 42, 43syl22anc 1229 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  ( S  _D  k )  e.  ( CC  ^pm  dom  F ) )
4525, 44syl5eqel 2549 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( k  e.  ( CC  ^pm  dom  F )  /\  n  e.  ( CC  ^pm  dom  F ) ) )  ->  (
k ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) n )  e.  ( CC 
^pm  dom  F ) )
461, 2, 16, 45seqf 12131 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  seq 0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) : NN0 --> ( CC 
^pm  dom  F ) )
4721dvnfval 22451 . . . 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 } ) ) )
4830, 47sylan 471 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  Dn F )  =  seq 0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) )
4948feq1d 5723 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  (
( S  Dn
F ) : NN0 --> ( CC  ^pm  dom  F )  <->  seq 0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) : NN0 --> ( CC 
^pm  dom  F ) ) )
5046, 49mpbird 232 1  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  Dn F ) : NN0 --> ( CC 
^pm  dom  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   {csn 4032   {cpr 4034    |-> cmpt 4515    X. cxp 5006   dom cdm 5008    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797    ^pm cpm 7439   CCcc 9507   RRcr 9508   0cc0 9509   NN0cn0 10816    seqcseq 12110    _D cdv 22393    Dncdvn 22394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-icc 11561  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-rest 14840  df-topn 14841  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cnp 19856  df-haus 19943  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-limc 22396  df-dv 22397  df-dvn 22398
This theorem is referenced by:  dvnf  22456  dvnbss  22457  dvnadd  22458
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