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Theorem dvnfre 21426
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 5691 . . . . . 6  |-  ( x  =  0  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  0 ) )
21dmeqd 5042 . . . . . 6  |-  ( x  =  0  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) ` 
0 ) )
31, 2feq12d 5548 . . . . 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 5691 . . . . . 6  |-  ( x  =  n  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  n ) )
65dmeqd 5042 . . . . . 6  |-  ( x  =  n  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  n ) )
75, 6feq12d 5548 . . . . 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 5691 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  ( n  +  1 ) ) )
109dmeqd 5042 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  ( n  +  1
) ) )
119, 10feq12d 5548 . . . . 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 5691 . . . . . 6  |-  ( x  =  N  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  N ) )
1413dmeqd 5042 . . . . . 6  |-  ( x  =  N  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  N ) )
1513, 14feq12d 5548 . . . . 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 457 . . . . 5  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F : A --> RR )
18 ax-resscn 9339 . . . . . . 7  |-  RR  C_  CC
19 fss 5567 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
2018, 19mpan2 671 . . . . . . . 8  |-  ( F : A --> RR  ->  F : A --> CC )
21 cnex 9363 . . . . . . . . 9  |-  CC  e.  _V
22 reex 9373 . . . . . . . . 9  |-  RR  e.  _V
23 elpm2r 7230 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  RR  e.  _V )  /\  ( F : A --> CC  /\  A  C_  RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
2421, 22, 23mpanl12 682 . . . . . . . 8  |-  ( ( F : A --> CC  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
2520, 24sylan 471 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
26 dvn0 21398 . . . . . . 7  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR ) )  ->  (
( RR  Dn
F ) `  0
)  =  F )
2718, 25, 26sylancr 663 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( RR  Dn F ) ` 
0 )  =  F )
2827dmeqd 5042 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  Dn F ) ` 
0 )  =  dom  F )
29 fdm 5563 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
3029adantr 465 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  F  =  A )
3128, 30eqtrd 2475 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  Dn F ) ` 
0 )  =  A )
3227, 31feq12d 5548 . . . . 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 756 . . . . . . . . 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 3983 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 755 . . . . . . . . . . . 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 21402 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 465 . . . . . . . . . . 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 3392 . . . . . . . . . 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 754 . . . . . . . . . 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 3366 . . . . . . . . 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 21425 . . . . . . . . 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 661 . . . . . . . 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 21399 . . . . . . . . . 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 1218 . . . . . . . . 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 5042 . . . . . . . . 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 5548 . . . . . . . 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 615 . . . . . 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 10738 . . 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 1184 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2972    C_ wss 3328   {cpr 3879   dom cdm 4840   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^pm cpm 7215   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285   NN0cn0 10579    _D cdv 21338    Dncdvn 21339
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fi 7661  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-icc 11307  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-plusg 14251  df-mulr 14252  df-starv 14253  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-rest 14361  df-topn 14362  df-topgen 14382  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-cncf 20454  df-limc 21341  df-dv 21342  df-dvn 21343
This theorem is referenced by:  taylthlem2  21839
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