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Theorem dvnff 19288
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  D n 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 10278 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10051 . . . 4  |-  0  e.  ZZ
32a1i 10 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  0  e.  ZZ )
4 fvconst2g 5743 . . . . 5  |-  ( ( F  e.  ( CC 
^pm  S )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  =  F )
54adantll 694 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  =  F )
6 dmexg 4955 . . . . . 6  |-  ( F  e.  ( CC  ^pm  S )  ->  dom  F  e. 
_V )
76ad2antlr 707 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  dom  F  e.  _V )
8 cnex 8834 . . . . . 6  |-  CC  e.  _V
98a1i 10 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  CC  e.  _V )
10 elpm2g 6803 . . . . . . . . 9  |-  ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  -> 
( F  e.  ( CC  ^pm  S )  <->  ( F : dom  F --> CC  /\  dom  F  C_  S ) ) )
118, 10mpan 651 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  ( F  e.  ( CC  ^pm  S
)  <->  ( F : dom  F --> CC  /\  dom  F 
C_  S ) ) )
1211biimpa 470 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  S )
)
1312simpld 445 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  F : dom  F --> CC )
1413adantr 451 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  F : dom  F --> CC )
15 fpmg 6809 . . . . 5  |-  ( ( dom  F  e.  _V  /\  CC  e.  _V  /\  F : dom  F --> CC )  ->  F  e.  ( CC  ^pm  dom  F ) )
167, 9, 14, 15syl3anc 1182 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  F  e.  ( CC  ^pm 
dom  F ) )
175, 16eqeltrd 2370 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  e.  ( CC  ^pm  dom  F ) )
18 vex 2804 . . . . . 6  |-  k  e. 
_V
19 vex 2804 . . . . . 6  |-  n  e. 
_V
2018, 19algrflem 6240 . . . . 5  |-  ( k ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) n )  =  ( ( x  e.  _V  |->  ( S  _D  x ) ) `
 k )
21 oveq2 5882 . . . . . . 7  |-  ( x  =  k  ->  ( S  _D  x )  =  ( S  _D  k
) )
22 eqid 2296 . . . . . . 7  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
23 ovex 5899 . . . . . . 7  |-  ( S  _D  k )  e. 
_V
2421, 22, 23fvmpt 5618 . . . . . 6  |-  ( k  e.  _V  ->  (
( x  e.  _V  |->  ( S  _D  x
) ) `  k
)  =  ( S  _D  k ) )
2518, 24ax-mp 8 . . . . 5  |-  ( ( x  e.  _V  |->  ( S  _D  x ) ) `  k )  =  ( S  _D  k )
2620, 25eqtri 2316 . . . 4  |-  ( k ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) n )  =  ( S  _D  k )
278a1i 10 . . . . 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 )
286ad2antlr 707 . . . . 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 )
29 dvfg 19272 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  k ) : dom  ( S  _D  k
) --> CC )
3029ad2antrr 706 . . . . 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 )
31 recnprss 19270 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
3231ad2antrr 706 . . . . . . 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 )
33 simprl 732 . . . . . . . . 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 ) )
34 elpm2g 6803 . . . . . . . . . 10  |-  ( ( CC  e.  _V  /\  dom  F  e.  _V )  ->  ( k  e.  ( CC  ^pm  dom  F )  <-> 
( k : dom  k
--> CC  /\  dom  k  C_ 
dom  F ) ) )
358, 28, 34sylancr 644 . . . . . . . . 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
) ) )
3633, 35mpbid 201 . . . . . . . 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 ) )
3736simpld 445 . . . . . . 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 )
3836simprd 449 . . . . . . . 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 )
3912simprd 449 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  dom  F 
C_  S )
4039adantr 451 . . . . . . . 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 )
4138, 40sstrd 3202 . . . . . . 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 )
4232, 37, 41dvbss 19267 . . . . . 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 )
4342, 38sstrd 3202 . . . . 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 )
44 elpm2r 6804 . . . . 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 ) )
4527, 28, 30, 43, 44syl22anc 1183 . . . 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 ) )
4626, 45syl5eqel 2380 . . 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 ) )
471, 3, 17, 46seqf 11083 . 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 ) )
4822dvnfval 19287 . . . 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 } ) ) )
4931, 48sylan 457 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  D n F )  =  seq  0 ( ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) ,  ( NN0  X.  { F } ) ) )
5049feq1d 5395 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  (
( S  D n F ) : NN0 --> ( CC  ^pm  dom  F )  <->  seq  0 ( ( ( x  e.  _V  |->  ( S  _D  x ) )  o.  1st ) ,  ( NN0  X.  { F } ) ) : NN0 --> ( CC 
^pm  dom  F ) ) )
5147, 50mpbird 223 1  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  D n F ) : NN0 --> ( CC 
^pm  dom  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653   {cpr 3654    e. cmpt 4093    X. cxp 4703   dom cdm 4705    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136    ^pm cpm 6789   CCcc 8751   RRcr 8752   0cc0 8753   NN0cn0 9981   ZZcz 10040    seq cseq 11062    _D cdv 19229    D ncdvn 19230
This theorem is referenced by:  dvnf  19292  dvnbss  19293  dvnadd  19294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cnp 16974  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-limc 19232  df-dv 19233  df-dvn 19234
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