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Theorem dvnff 21419
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 10916 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 10679 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  0  e.  ZZ )
3 fvconst2g 5952 . . . . 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 6530 . . . . . 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 9384 . . . . . 6  |-  CC  e.  _V
87a1i 11 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  CC  e.  _V )
9 elpm2g 7250 . . . . . . . . 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 7259 . . . . 5  |-  ( ( dom  F  e.  _V  /\  CC  e.  _V  /\  F : dom  F --> CC )  ->  F  e.  ( CC  ^pm  dom  F ) )
156, 8, 13, 14syl3anc 1218 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  ->  F  e.  ( CC  ^pm 
dom  F ) )
164, 15eqeltrd 2517 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  k  e.  NN0 )  -> 
( ( NN0  X.  { F } ) `  k )  e.  ( CC  ^pm  dom  F ) )
17 vex 2996 . . . . . 6  |-  k  e. 
_V
18 vex 2996 . . . . . 6  |-  n  e. 
_V
1917, 18algrflem 6702 . . . . 5  |-  ( k ( ( x  e. 
_V  |->  ( S  _D  x ) )  o. 
1st ) n )  =  ( ( x  e.  _V  |->  ( S  _D  x ) ) `
 k )
20 oveq2 6120 . . . . . . 7  |-  ( x  =  k  ->  ( S  _D  x )  =  ( S  _D  k
) )
21 eqid 2443 . . . . . . 7  |-  ( x  e.  _V  |->  ( S  _D  x ) )  =  ( x  e. 
_V  |->  ( S  _D  x ) )
22 ovex 6137 . . . . . . 7  |-  ( S  _D  k )  e. 
_V
2320, 21, 22fvmpt 5795 . . . . . 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 2463 . . . 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 21403 . . . . . 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 21401 . . . . . . . 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 755 . . . . . . . . 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 7250 . . . . . . . . . 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 3387 . . . . . . 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 21398 . . . . . 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 3387 . . . . 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 7251 . . . . 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 1219 . . . 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 2527 . . 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 11848 . 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 21418 . . . 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 5567 . 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 1369    e. wcel 1756   _Vcvv 2993    C_ wss 3349   {csn 3898   {cpr 3900    e. cmpt 4371    X. cxp 4859   dom cdm 4861    o. ccom 4865   -->wf 5435   ` cfv 5439  (class class class)co 6112   1stc1st 6596    ^pm cpm 7236   CCcc 9301   RRcr 9302   0cc0 9303   NN0cn0 10600    seqcseq 11827    _D cdv 21360    Dncdvn 21361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fi 7682  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-icc 11328  df-fz 11459  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-rest 14382  df-topn 14383  df-topgen 14403  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cnp 18854  df-haus 18941  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-xms 19917  df-ms 19918  df-limc 21363  df-dv 21364  df-dvn 21365
This theorem is referenced by:  dvnf  21423  dvnbss  21424  dvnadd  21425
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