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Theorem dvnff 19676
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 10452 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10225 . . . 4  |-  0  e.  ZZ
32a1i 11 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  0  e.  ZZ )
4 fvconst2g 5884 . . . . 5  |-  ( ( F  e.  ( CC 
^pm  S )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  =  F )
54adantll 695 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  =  F )
6 dmexg 5070 . . . . . 6  |-  ( F  e.  ( CC  ^pm  S )  ->  dom  F  e. 
_V )
76ad2antlr 708 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  dom  F  e.  _V )
8 cnex 9004 . . . . . 6  |-  CC  e.  _V
98a1i 11 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  CC  e.  _V )
10 elpm2g 6969 . . . . . . . . 9  |-  ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  -> 
( F  e.  ( CC  ^pm  S )  <->  ( F : dom  F --> CC  /\  dom  F  C_  S ) ) )
118, 10mpan 652 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  ( F  e.  ( CC  ^pm  S
)  <->  ( F : dom  F --> CC  /\  dom  F 
C_  S ) ) )
1211biimpa 471 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  S )
)
1312simpld 446 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  F : dom  F --> CC )
1413adantr 452 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  F : dom  F --> CC )
15 fpmg 6975 . . . . 5  |-  ( ( dom  F  e.  _V  /\  CC  e.  _V  /\  F : dom  F --> CC )  ->  F  e.  ( CC  ^pm  dom  F ) )
167, 9, 14, 15syl3anc 1184 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  F  e.  ( CC  ^pm 
dom  F ) )
175, 16eqeltrd 2461 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  e.  ( CC  ^pm  dom  F ) )
18 vex 2902 . . . . . 6  |-  k  e. 
_V
19 vex 2902 . . . . . 6  |-  n  e. 
_V
2018, 19algrflem 6391 . . . . 5  |-  ( k ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) n )  =  ( ( x  e.  _V  |->  ( S  _D  x ) ) `
 k )
21 oveq2 6028 . . . . . . 7  |-  ( x  =  k  ->  ( S  _D  x )  =  ( S  _D  k
) )
22 eqid 2387 . . . . . . 7  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
23 ovex 6045 . . . . . . 7  |-  ( S  _D  k )  e. 
_V
2421, 22, 23fvmpt 5745 . . . . . 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 2407 . . . 4  |-  ( k ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) n )  =  ( S  _D  k )
278a1i 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 )
286ad2antlr 708 . . . . 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 19660 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  k ) : dom  ( S  _D  k
) --> CC )
3029ad2antrr 707 . . . . 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 19658 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
3231ad2antrr 707 . . . . . . 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 733 . . . . . . . . 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 6969 . . . . . . . . . 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 645 . . . . . . . . 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 202 . . . . . . . 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 446 . . . . . . 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 450 . . . . . . . 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 450 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  dom  F 
C_  S )
4039adantr 452 . . . . . . . 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 3301 . . . . . . 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 19655 . . . . . 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 3301 . . . . 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 6970 . . . . 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 1185 . . . 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 2471 . . 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 11271 . 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 19675 . . . 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 458 . . 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 5520 . 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 224 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263   {csn 3757   {cpr 3758    e. cmpt 4207    X. cxp 4816   dom cdm 4818    o. ccom 4822   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286    ^pm cpm 6955   CCcc 8921   RRcr 8922   0cc0 8923   NN0cn0 10153   ZZcz 10214    seq cseq 11250    _D cdv 19617    D ncdvn 19618
This theorem is referenced by:  dvnf  19680  dvnbss  19681  dvnadd  19682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-icc 10855  df-fz 10976  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-plusg 13469  df-mulr 13470  df-starv 13471  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-rest 13577  df-topn 13578  df-topgen 13594  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cnp 17214  df-haus 17301  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-limc 19620  df-dv 19621  df-dvn 19622
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