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Theorem dvnfre 22200
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 5871 . . . . . 6  |-  ( x  =  0  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  0 ) )
21dmeqd 5210 . . . . . 6  |-  ( x  =  0  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) ` 
0 ) )
31, 2feq12d 5725 . . . . 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 5871 . . . . . 6  |-  ( x  =  n  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  n ) )
65dmeqd 5210 . . . . . 6  |-  ( x  =  n  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  n ) )
75, 6feq12d 5725 . . . . 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 5871 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  ( n  +  1 ) ) )
109dmeqd 5210 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  ( n  +  1
) ) )
119, 10feq12d 5725 . . . . 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 5871 . . . . . 6  |-  ( x  =  N  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  N ) )
1413dmeqd 5210 . . . . . 6  |-  ( x  =  N  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  N ) )
1513, 14feq12d 5725 . . . . 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 9559 . . . . . . 7  |-  RR  C_  CC
19 fss 5744 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
2018, 19mpan2 671 . . . . . . . 8  |-  ( F : A --> RR  ->  F : A --> CC )
21 cnex 9583 . . . . . . . . 9  |-  CC  e.  _V
22 reex 9593 . . . . . . . . 9  |-  RR  e.  _V
23 elpm2r 7446 . . . . . . . . 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 22172 . . . . . . 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 5210 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  Dn F ) ` 
0 )  =  dom  F )
29 fdm 5740 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
3029adantr 465 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  F  =  A )
3128, 30eqtrd 2508 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  Dn F ) ` 
0 )  =  A )
3227, 31feq12d 5725 . . . . 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 4140 . . . . . . . . . . . . 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 22176 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 3545 . . . . . . . . . 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 3519 . . . . . . . . 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 22199 . . . . . . . . 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 22173 . . . . . . . . . 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 1228 . . . . . . . . 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 5210 . . . . . . . . 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 5725 . . . . . . . 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 10967 . . 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 1193 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   {cpr 4034   dom cdm 5004   -->wf 5589   ` cfv 5593  (class class class)co 6294    ^pm cpm 7431   CCcc 9500   RRcr 9501   0cc0 9502   1c1 9503    + caddc 9505   NN0cn0 10805    _D cdv 22112    Dncdvn 22113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fi 7881  df-sup 7911  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ioo 11543  df-icc 11546  df-fz 11683  df-seq 12086  df-exp 12145  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-plusg 14580  df-mulr 14581  df-starv 14582  df-tset 14586  df-ple 14587  df-ds 14589  df-unif 14590  df-rest 14690  df-topn 14691  df-topgen 14711  df-psmet 18258  df-xmet 18259  df-met 18260  df-bl 18261  df-mopn 18262  df-fbas 18263  df-fg 18264  df-cnfld 18268  df-top 19245  df-bases 19247  df-topon 19248  df-topsp 19249  df-cld 19365  df-ntr 19366  df-cls 19367  df-nei 19444  df-lp 19482  df-perf 19483  df-cn 19573  df-cnp 19574  df-haus 19661  df-fil 20192  df-fm 20284  df-flim 20285  df-flf 20286  df-xms 20668  df-ms 20669  df-cncf 21227  df-limc 22115  df-dv 22116  df-dvn 22117
This theorem is referenced by:  taylthlem2  22613
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