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Theorem dvnfre 21326
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  Dn
F ) `  N
) : dom  (
( RR  Dn
F ) `  N
) --> RR )

Proof of Theorem dvnfre
Dummy variables  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5688 . . . . . 6  |-  ( x  =  0  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  0 ) )
21dmeqd 5038 . . . . . 6  |-  ( x  =  0  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) ` 
0 ) )
31, 2feq12d 5545 . . . . 5  |-  ( x  =  0  ->  (
( ( RR  Dn F ) `  x ) : dom  ( ( RR  Dn F ) `  x ) --> RR  <->  ( ( RR  Dn F ) `
 0 ) : dom  ( ( RR  Dn F ) `
 0 ) --> RR ) )
43imbi2d 316 . . . 4  |-  ( x  =  0  ->  (
( ( F : A
--> RR  /\  A  C_  RR )  ->  ( ( RR  Dn F ) `  x ) : dom  ( ( RR  Dn F ) `  x ) --> RR )  <->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) ` 
0 ) : dom  ( ( RR  Dn F ) ` 
0 ) --> RR ) ) )
5 fveq2 5688 . . . . . 6  |-  ( x  =  n  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  n ) )
65dmeqd 5038 . . . . . 6  |-  ( x  =  n  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  n ) )
75, 6feq12d 5545 . . . . 5  |-  ( x  =  n  ->  (
( ( RR  Dn F ) `  x ) : dom  ( ( RR  Dn F ) `  x ) --> RR  <->  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )
87imbi2d 316 . . . 4  |-  ( x  =  n  ->  (
( ( F : A
--> RR  /\  A  C_  RR )  ->  ( ( RR  Dn F ) `  x ) : dom  ( ( RR  Dn F ) `  x ) --> RR )  <->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) `  n ) : dom  ( ( RR  Dn F ) `  n ) --> RR ) ) )
9 fveq2 5688 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  ( n  +  1 ) ) )
109dmeqd 5038 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  ( n  +  1
) ) )
119, 10feq12d 5545 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( ( RR  Dn F ) `  x ) : dom  ( ( RR  Dn F ) `  x ) --> RR  <->  ( ( RR  Dn F ) `
 ( n  + 
1 ) ) : dom  ( ( RR  Dn F ) `
 ( n  + 
1 ) ) --> RR ) )
1211imbi2d 316 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ( F : A
--> RR  /\  A  C_  RR )  ->  ( ( RR  Dn F ) `  x ) : dom  ( ( RR  Dn F ) `  x ) --> RR )  <->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) `  ( n  +  1
) ) : dom  ( ( RR  Dn F ) `  ( n  +  1
) ) --> RR ) ) )
13 fveq2 5688 . . . . . 6  |-  ( x  =  N  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  N ) )
1413dmeqd 5038 . . . . . 6  |-  ( x  =  N  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  N ) )
1513, 14feq12d 5545 . . . . 5  |-  ( x  =  N  ->  (
( ( RR  Dn F ) `  x ) : dom  ( ( RR  Dn F ) `  x ) --> RR  <->  ( ( RR  Dn F ) `
 N ) : dom  ( ( RR  Dn F ) `
 N ) --> RR ) )
1615imbi2d 316 . . . 4  |-  ( x  =  N  ->  (
( ( F : A
--> RR  /\  A  C_  RR )  ->  ( ( RR  Dn F ) `  x ) : dom  ( ( RR  Dn F ) `  x ) --> RR )  <->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) `  N ) : dom  ( ( RR  Dn F ) `  N ) --> RR ) ) )
17 simpl 454 . . . . 5  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F : A --> RR )
18 ax-resscn 9335 . . . . . . 7  |-  RR  C_  CC
19 fss 5564 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
2018, 19mpan2 666 . . . . . . . 8  |-  ( F : A --> RR  ->  F : A --> CC )
21 cnex 9359 . . . . . . . . 9  |-  CC  e.  _V
22 reex 9369 . . . . . . . . 9  |-  RR  e.  _V
23 elpm2r 7226 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  RR  e.  _V )  /\  ( F : A --> CC  /\  A  C_  RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
2421, 22, 23mpanl12 677 . . . . . . . 8  |-  ( ( F : A --> CC  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
2520, 24sylan 468 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
26 dvn0 21298 . . . . . . 7  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR ) )  ->  (
( RR  Dn
F ) `  0
)  =  F )
2718, 25, 26sylancr 658 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) ` 
0 )  =  F )
2827dmeqd 5038 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  Dn F ) ` 
0 )  =  dom  F )
29 fdm 5560 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
3029adantr 462 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  F  =  A )
3128, 30eqtrd 2473 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  Dn F ) ` 
0 )  =  A )
3227, 31feq12d 5545 . . . . 5  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( ( RR  Dn F ) `
 0 ) : dom  ( ( RR  Dn F ) `
 0 ) --> RR  <->  F : A --> RR ) )
3317, 32mpbird 232 . . . 4  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) ` 
0 ) : dom  ( ( RR  Dn F ) ` 
0 ) --> RR )
34 simprr 751 . . . . . . . . 9  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  (
( RR  Dn
F ) `  n
) : dom  (
( RR  Dn
F ) `  n
) --> RR )
3522prid1 3980 . . . . . . . . . . . . 13  |-  RR  e.  { RR ,  CC }
3635a1i 11 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  RR  e.  { RR ,  CC } )
3725adantr 462 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
38 simprl 750 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  n  e.  NN0 )
39 dvnbss 21302 . . . . . . . . . . . 12  |-  ( ( RR  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  RR )  /\  n  e.  NN0 )  ->  dom  ( ( RR  Dn F ) `
 n )  C_  dom  F )
4036, 37, 38, 39syl3anc 1213 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  dom  ( ( RR  Dn F ) `  n )  C_  dom  F )
4130adantr 462 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  dom  F  =  A )
4240, 41sseqtrd 3389 . . . . . . . . . 10  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  dom  ( ( RR  Dn F ) `  n )  C_  A
)
43 simplr 749 . . . . . . . . . 10  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  A  C_  RR )
4442, 43sstrd 3363 . . . . . . . . 9  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  dom  ( ( RR  Dn F ) `  n )  C_  RR )
45 dvfre 21325 . . . . . . . . 9  |-  ( ( ( ( RR  Dn F ) `  n ) : dom  ( ( RR  Dn F ) `  n ) --> RR  /\  dom  ( ( RR  Dn F ) `  n )  C_  RR )  ->  ( RR  _D  ( ( RR  Dn F ) `  n ) ) : dom  ( RR  _D  ( ( RR  Dn F ) `  n ) ) --> RR )
4634, 44, 45syl2anc 656 . . . . . . . 8  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  ( RR  _D  ( ( RR  Dn F ) `
 n ) ) : dom  ( RR 
_D  ( ( RR  Dn F ) `
 n ) ) --> RR )
4718a1i 11 . . . . . . . . . 10  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  RR  C_  CC )
48 dvnp1 21299 . . . . . . . . . 10  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR )  /\  n  e.  NN0 )  ->  ( ( RR  Dn F ) `
 ( n  + 
1 ) )  =  ( RR  _D  (
( RR  Dn
F ) `  n
) ) )
4947, 37, 38, 48syl3anc 1213 . . . . . . . . 9  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  (
( RR  Dn
F ) `  (
n  +  1 ) )  =  ( RR 
_D  ( ( RR  Dn F ) `
 n ) ) )
5049dmeqd 5038 . . . . . . . . 9  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  dom  ( ( RR  Dn F ) `  ( n  +  1
) )  =  dom  ( RR  _D  (
( RR  Dn
F ) `  n
) ) )
5149, 50feq12d 5545 . . . . . . . 8  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  (
( ( RR  Dn F ) `  ( n  +  1
) ) : dom  ( ( RR  Dn F ) `  ( n  +  1
) ) --> RR  <->  ( RR  _D  ( ( RR  Dn F ) `  n ) ) : dom  ( RR  _D  ( ( RR  Dn F ) `  n ) ) --> RR ) )
5246, 51mpbird 232 . . . . . . 7  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( n  e. 
NN0  /\  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR ) )  ->  (
( RR  Dn
F ) `  (
n  +  1 ) ) : dom  (
( RR  Dn
F ) `  (
n  +  1 ) ) --> RR )
5352expr 612 . . . . . 6  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  n  e.  NN0 )  ->  ( ( ( RR  Dn F ) `  n ) : dom  ( ( RR  Dn F ) `  n ) --> RR  ->  ( ( RR  Dn F ) `
 ( n  + 
1 ) ) : dom  ( ( RR  Dn F ) `
 ( n  + 
1 ) ) --> RR ) )
5453expcom 435 . . . . 5  |-  ( n  e.  NN0  ->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR 
->  ( ( RR  Dn F ) `  ( n  +  1
) ) : dom  ( ( RR  Dn F ) `  ( n  +  1
) ) --> RR ) ) )
5554a2d 26 . . . 4  |-  ( n  e.  NN0  ->  ( ( ( F : A --> RR  /\  A  C_  RR )  ->  ( ( RR  Dn F ) `
 n ) : dom  ( ( RR  Dn F ) `
 n ) --> RR )  ->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) `  ( n  +  1
) ) : dom  ( ( RR  Dn F ) `  ( n  +  1
) ) --> RR ) ) )
564, 8, 12, 16, 33, 55nn0ind 10734 . . 3  |-  ( N  e.  NN0  ->  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) `  N ) : dom  ( ( RR  Dn F ) `  N ) --> RR ) )
5756com12 31 . 2  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( N  e.  NN0  ->  ( ( RR  Dn F ) `  N ) : dom  ( ( RR  Dn F ) `  N ) --> RR ) )
58573impia 1179 1  |-  ( ( F : A --> RR  /\  A  C_  RR  /\  N  e.  NN0 )  ->  (
( RR  Dn
F ) `  N
) : dom  (
( RR  Dn
F ) `  N
) --> RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   {cpr 3876   dom cdm 4836   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^pm cpm 7211   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281   NN0cn0 10575    _D cdv 21238    Dncdvn 21239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-icc 11303  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-rest 14357  df-topn 14358  df-topgen 14378  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-haus 18819  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-xms 19795  df-ms 19796  df-cncf 20354  df-limc 21241  df-dv 21242  df-dvn 21243
This theorem is referenced by:  taylthlem2  21782
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