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Theorem dvnff 19272
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 10262 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10035 . . . 4  |-  0  e.  ZZ
32a1i 10 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  0  e.  ZZ )
4 fvconst2g 5727 . . . . 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 4939 . . . . . 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 8818 . . . . . 6  |-  CC  e.  _V
98a1i 10 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  CC  e.  _V )
10 elpm2g 6787 . . . . . . . . 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 6793 . . . . 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 2357 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  e.  ( CC  ^pm  dom  F ) )
18 vex 2791 . . . . . 6  |-  k  e. 
_V
19 vex 2791 . . . . . 6  |-  n  e. 
_V
2018, 19algrflem 6224 . . . . 5  |-  ( k ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) n )  =  ( ( x  e.  _V  |->  ( S  _D  x ) ) `
 k )
21 oveq2 5866 . . . . . . 7  |-  ( x  =  k  ->  ( S  _D  x )  =  ( S  _D  k
) )
22 eqid 2283 . . . . . . 7  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
23 ovex 5883 . . . . . . 7  |-  ( S  _D  k )  e. 
_V
2421, 22, 23fvmpt 5602 . . . . . 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 2303 . . . 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 19256 . . . . . 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 19254 . . . . . . . 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 6787 . . . . . . . . . 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 3189 . . . . . . 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 19251 . . . . . 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 3189 . . . . 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 6788 . . . . 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 2367 . . 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 11067 . 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 19271 . . . 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 5379 . 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640   {cpr 3641    e. cmpt 4077    X. cxp 4687   dom cdm 4689    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120    ^pm cpm 6773   CCcc 8735   RRcr 8736   0cc0 8737   NN0cn0 9965   ZZcz 10024    seq cseq 11046    _D cdv 19213    D ncdvn 19214
This theorem is referenced by:  dvnf  19276  dvnbss  19277  dvnadd  19278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cnp 16958  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-limc 19216  df-dv 19217  df-dvn 19218
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