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Theorem dvnfre 19399
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 5605 . . . . . 6  |-  ( x  =  0  ->  (
( RR  D n F ) `  x
)  =  ( ( RR  D n F ) `  0 ) )
21dmeqd 4960 . . . . . 6  |-  ( x  =  0  ->  dom  ( ( RR  D n F ) `  x
)  =  dom  (
( RR  D n F ) `  0
) )
31, 2feq12d 5460 . . . . 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 5605 . . . . . 6  |-  ( x  =  n  ->  (
( RR  D n F ) `  x
)  =  ( ( RR  D n F ) `  n ) )
65dmeqd 4960 . . . . . 6  |-  ( x  =  n  ->  dom  ( ( RR  D n F ) `  x
)  =  dom  (
( RR  D n F ) `  n
) )
75, 6feq12d 5460 . . . . 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 5605 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( RR  D n F ) `  x
)  =  ( ( RR  D n F ) `  ( n  +  1 ) ) )
109dmeqd 4960 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( RR  D n F ) `  x
)  =  dom  (
( RR  D n F ) `  (
n  +  1 ) ) )
119, 10feq12d 5460 . . . . 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 5605 . . . . . 6  |-  ( x  =  N  ->  (
( RR  D n F ) `  x
)  =  ( ( RR  D n F ) `  N ) )
1413dmeqd 4960 . . . . . 6  |-  ( x  =  N  ->  dom  ( ( RR  D n F ) `  x
)  =  dom  (
( RR  D n F ) `  N
) )
1513, 14feq12d 5460 . . . . 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 8881 . . . . . . 7  |-  RR  C_  CC
19 fss 5477 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
2018, 19mpan2 652 . . . . . . . 8  |-  ( F : A --> RR  ->  F : A --> CC )
21 cnex 8905 . . . . . . . . 9  |-  CC  e.  _V
22 reex 8915 . . . . . . . . 9  |-  RR  e.  _V
23 elpm2r 6873 . . . . . . . . 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 19371 . . . . . . 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 4960 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  D n F ) `  0
)  =  dom  F
)
29 fdm 5473 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
3029adantr 451 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  F  =  A )
3128, 30eqtrd 2390 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  D n F ) `  0
)  =  A )
3227, 31feq12d 5460 . . . . 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 3810 . . . . . . . . . . . . 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 19375 . . . . . . . . . . . 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 3290 . . . . . . . . . 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 3265 . . . . . . . . 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 19398 . . . . . . . . 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 19372 . . . . . . . . . 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 4960 . . . . . . . . 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 5460 . . . . . . . 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 10197 . . 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 1642    e. wcel 1710   _Vcvv 2864    C_ wss 3228   {cpr 3717   dom cdm 4768   -->wf 5330   ` cfv 5334  (class class class)co 5942    ^pm cpm 6858   CCcc 8822   RRcr 8823   0cc0 8824   1c1 8825    + caddc 8827   NN0cn0 10054    _D cdv 19311    D ncdvn 19312
This theorem is referenced by:  taylthlem2  19851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-icc 10752  df-fz 10872  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-plusg 13312  df-mulr 13313  df-starv 13314  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-rest 13420  df-topn 13421  df-topgen 13437  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-cnfld 16477  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lp 16968  df-perf 16969  df-cn 17057  df-cnp 17058  df-haus 17143  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-xms 17981  df-ms 17982  df-cncf 18479  df-limc 19314  df-dv 19315  df-dvn 19316
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