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Theorem dvnfre 19301
Description: The  N-th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)
Assertion
Ref Expression
dvnfre  |-  ( ( F : A --> RR  /\  A  C_  RR  /\  N  e.  NN0 )  ->  (
( RR  D n F ) `  N
) : dom  (
( RR  D n F ) `  N
) --> RR )

Proof of Theorem dvnfre
Dummy variables  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . . 6  |-  ( x  =  0  ->  (
( RR  D n F ) `  x
)  =  ( ( RR  D n F ) `  0 ) )
21dmeqd 4881 . . . . . 6  |-  ( x  =  0  ->  dom  ( ( RR  D n F ) `  x
)  =  dom  (
( RR  D n F ) `  0
) )
31, 2feq12d 5381 . . . . 5  |-  ( x  =  0  ->  (
( ( RR  D n F ) `  x
) : dom  (
( RR  D n F ) `  x
) --> RR  <->  ( ( RR  D n F ) `
 0 ) : dom  ( ( RR  D n F ) `
 0 ) --> RR ) )
43imbi2d 307 . . . 4  |-  ( x  =  0  ->  (
( ( F : A
--> RR  /\  A  C_  RR )  ->  ( ( RR  D n F ) `  x ) : dom  ( ( RR  D n F ) `  x ) --> RR )  <->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  D n F ) `  0
) : dom  (
( RR  D n F ) `  0
) --> RR ) ) )
5 fveq2 5525 . . . . . 6  |-  ( x  =  n  ->  (
( RR  D n F ) `  x
)  =  ( ( RR  D n F ) `  n ) )
65dmeqd 4881 . . . . . 6  |-  ( x  =  n  ->  dom  ( ( RR  D n F ) `  x
)  =  dom  (
( RR  D n F ) `  n
) )
75, 6feq12d 5381 . . . . 5  |-  ( x  =  n  ->  (
( ( RR  D n F ) `  x
) : dom  (
( RR  D n F ) `  x
) --> RR  <->  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )
87imbi2d 307 . . . 4  |-  ( x  =  n  ->  (
( ( F : A
--> RR  /\  A  C_  RR )  ->  ( ( RR  D n F ) `  x ) : dom  ( ( RR  D n F ) `  x ) --> RR )  <->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  D n F ) `  n
) : dom  (
( RR  D n F ) `  n
) --> RR ) ) )
9 fveq2 5525 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( RR  D n F ) `  x
)  =  ( ( RR  D n F ) `  ( n  +  1 ) ) )
109dmeqd 4881 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( RR  D n F ) `  x
)  =  dom  (
( RR  D n F ) `  (
n  +  1 ) ) )
119, 10feq12d 5381 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( ( RR  D n F ) `  x
) : dom  (
( RR  D n F ) `  x
) --> RR  <->  ( ( RR  D n F ) `
 ( n  + 
1 ) ) : dom  ( ( RR  D n F ) `
 ( n  + 
1 ) ) --> RR ) )
1211imbi2d 307 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ( F : A
--> RR  /\  A  C_  RR )  ->  ( ( RR  D n F ) `  x ) : dom  ( ( RR  D n F ) `  x ) --> RR )  <->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  D n F ) `  (
n  +  1 ) ) : dom  (
( RR  D n F ) `  (
n  +  1 ) ) --> RR ) ) )
13 fveq2 5525 . . . . . 6  |-  ( x  =  N  ->  (
( RR  D n F ) `  x
)  =  ( ( RR  D n F ) `  N ) )
1413dmeqd 4881 . . . . . 6  |-  ( x  =  N  ->  dom  ( ( RR  D n F ) `  x
)  =  dom  (
( RR  D n F ) `  N
) )
1513, 14feq12d 5381 . . . . 5  |-  ( x  =  N  ->  (
( ( RR  D n F ) `  x
) : dom  (
( RR  D n F ) `  x
) --> RR  <->  ( ( RR  D n F ) `
 N ) : dom  ( ( RR  D n F ) `
 N ) --> RR ) )
1615imbi2d 307 . . . 4  |-  ( x  =  N  ->  (
( ( F : A
--> RR  /\  A  C_  RR )  ->  ( ( RR  D n F ) `  x ) : dom  ( ( RR  D n F ) `  x ) --> RR )  <->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  D n F ) `  N
) : dom  (
( RR  D n F ) `  N
) --> RR ) ) )
17 simpl 443 . . . . 5  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F : A --> RR )
18 ax-resscn 8794 . . . . . . 7  |-  RR  C_  CC
19 fss 5397 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
2018, 19mpan2 652 . . . . . . . 8  |-  ( F : A --> RR  ->  F : A --> CC )
21 cnex 8818 . . . . . . . . 9  |-  CC  e.  _V
22 reex 8828 . . . . . . . . 9  |-  RR  e.  _V
23 elpm2r 6788 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  RR  e.  _V )  /\  ( F : A --> CC  /\  A  C_  RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
2421, 22, 23mpanl12 663 . . . . . . . 8  |-  ( ( F : A --> CC  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
2520, 24sylan 457 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
26 dvn0 19273 . . . . . . 7  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR ) )  ->  (
( RR  D n F ) `  0
)  =  F )
2718, 25, 26sylancr 644 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  D n F ) `  0
)  =  F )
2827dmeqd 4881 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  D n F ) `  0
)  =  dom  F
)
29 fdm 5393 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
3029adantr 451 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  F  =  A )
3128, 30eqtrd 2315 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  D n F ) `  0
)  =  A )
3227, 31feq12d 5381 . . . . 5  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( ( RR  D n F ) `
 0 ) : dom  ( ( RR  D n F ) `
 0 ) --> RR  <->  F : A --> RR ) )
3317, 32mpbird 223 . . . 4  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  D n F ) `  0
) : dom  (
( RR  D n F ) `  0
) --> RR )
34 simprr 733 . . . . . . . . 9  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  (
( RR  D n F ) `  n
) : dom  (
( RR  D n F ) `  n
) --> RR )
3522prid1 3734 . . . . . . . . . . . . 13  |-  RR  e.  { RR ,  CC }
3635a1i 10 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  RR  e.  { RR ,  CC } )
3725adantr 451 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
38 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  n  e.  NN0 )
39 dvnbss 19277 . . . . . . . . . . . 12  |-  ( ( RR  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  RR )  /\  n  e.  NN0 )  ->  dom  ( ( RR  D n F ) `
 n )  C_  dom  F )
4036, 37, 38, 39syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  dom  ( ( RR  D n F ) `  n
)  C_  dom  F )
4130adantr 451 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  dom  F  =  A )
4240, 41sseqtrd 3214 . . . . . . . . . 10  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  dom  ( ( RR  D n F ) `  n
)  C_  A )
43 simplr 731 . . . . . . . . . 10  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  A  C_  RR )
4442, 43sstrd 3189 . . . . . . . . 9  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  dom  ( ( RR  D n F ) `  n
)  C_  RR )
45 dvfre 19300 . . . . . . . . 9  |-  ( ( ( ( RR  D n F ) `  n
) : dom  (
( RR  D n F ) `  n
) --> RR  /\  dom  ( ( RR  D n F ) `  n
)  C_  RR )  ->  ( RR  _D  (
( RR  D n F ) `  n
) ) : dom  ( RR  _D  (
( RR  D n F ) `  n
) ) --> RR )
4634, 44, 45syl2anc 642 . . . . . . . 8  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  ( RR  _D  ( ( RR  D n F ) `
 n ) ) : dom  ( RR 
_D  ( ( RR  D n F ) `
 n ) ) --> RR )
4718a1i 10 . . . . . . . . . 10  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  RR  C_  CC )
48 dvnp1 19274 . . . . . . . . . 10  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR )  /\  n  e.  NN0 )  ->  ( ( RR  D n F ) `
 ( n  + 
1 ) )  =  ( RR  _D  (
( RR  D n F ) `  n
) ) )
4947, 37, 38, 48syl3anc 1182 . . . . . . . . 9  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  (
( RR  D n F ) `  (
n  +  1 ) )  =  ( RR 
_D  ( ( RR  D n F ) `
 n ) ) )
5049dmeqd 4881 . . . . . . . . 9  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  dom  ( ( RR  D n F ) `  (
n  +  1 ) )  =  dom  ( RR  _D  ( ( RR  D n F ) `
 n ) ) )
5149, 50feq12d 5381 . . . . . . . 8  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  (
( ( RR  D n F ) `  (
n  +  1 ) ) : dom  (
( RR  D n F ) `  (
n  +  1 ) ) --> RR  <->  ( RR  _D  ( ( RR  D n F ) `  n
) ) : dom  ( RR  _D  (
( RR  D n F ) `  n
) ) --> RR ) )
5246, 51mpbird 223 . . . . . . 7  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR ) )  ->  (
( RR  D n F ) `  (
n  +  1 ) ) : dom  (
( RR  D n F ) `  (
n  +  1 ) ) --> RR )
5352expr 598 . . . . . 6  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  n  e.  NN0 )  ->  ( ( ( RR  D n F ) `  n ) : dom  ( ( RR  D n F ) `  n ) --> RR  ->  ( ( RR  D n F ) `
 ( n  + 
1 ) ) : dom  ( ( RR  D n F ) `
 ( n  + 
1 ) ) --> RR ) )
5453expcom 424 . . . . 5  |-  ( n  e.  NN0  ->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR 
->  ( ( RR  D n F ) `  (
n  +  1 ) ) : dom  (
( RR  D n F ) `  (
n  +  1 ) ) --> RR ) ) )
5554a2d 23 . . . 4  |-  ( n  e.  NN0  ->  ( ( ( F : A --> RR  /\  A  C_  RR )  ->  ( ( RR  D n F ) `
 n ) : dom  ( ( RR  D n F ) `
 n ) --> RR )  ->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  D n F ) `  (
n  +  1 ) ) : dom  (
( RR  D n F ) `  (
n  +  1 ) ) --> RR ) ) )
564, 8, 12, 16, 33, 55nn0ind 10108 . . 3  |-  ( N  e.  NN0  ->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  D n F ) `  N
) : dom  (
( RR  D n F ) `  N
) --> RR ) )
5756com12 27 . 2  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( N  e.  NN0  ->  ( ( RR  D n F ) `  N
) : dom  (
( RR  D n F ) `  N
) --> RR ) )
58573impia 1148 1  |-  ( ( F : A --> RR  /\  A  C_  RR  /\  N  e.  NN0 )  ->  (
( RR  D n F ) `  N
) : dom  (
( RR  D n F ) `  N
) --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {cpr 3641   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   NN0cn0 9965    _D cdv 19213    D ncdvn 19214
This theorem is referenced by:  taylthlem2  19753
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-ioo 10660  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-cn 16957  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-cncf 18382  df-limc 19216  df-dv 19217  df-dvn 19218
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