MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvnfre Structured version   Unicode version

Theorem dvnfre 22228
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 5856 . . . . . 6  |-  ( x  =  0  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  0 ) )
21dmeqd 5195 . . . . . 6  |-  ( x  =  0  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) ` 
0 ) )
31, 2feq12d 5710 . . . . 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 5856 . . . . . 6  |-  ( x  =  n  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  n ) )
65dmeqd 5195 . . . . . 6  |-  ( x  =  n  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  n ) )
75, 6feq12d 5710 . . . . 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 5856 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  ( n  +  1 ) ) )
109dmeqd 5195 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  ( n  +  1
) ) )
119, 10feq12d 5710 . . . . 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 5856 . . . . . 6  |-  ( x  =  N  ->  (
( RR  Dn
F ) `  x
)  =  ( ( RR  Dn F ) `  N ) )
1413dmeqd 5195 . . . . . 6  |-  ( x  =  N  ->  dom  ( ( RR  Dn F ) `  x )  =  dom  ( ( RR  Dn F ) `  N ) )
1513, 14feq12d 5710 . . . . 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 9552 . . . . . . 7  |-  RR  C_  CC
19 fss 5729 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
2018, 19mpan2 671 . . . . . . . 8  |-  ( F : A --> RR  ->  F : A --> CC )
21 cnex 9576 . . . . . . . . 9  |-  CC  e.  _V
22 reex 9586 . . . . . . . . 9  |-  RR  e.  _V
23 elpm2r 7438 . . . . . . . . 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 22200 . . . . . . 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 5195 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  Dn F ) ` 
0 )  =  dom  F )
29 fdm 5725 . . . . . . . 8  |-  ( F : A --> RR  ->  dom 
F  =  A )
3029adantr 465 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  F  =  A )
3128, 30eqtrd 2484 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  dom  ( ( RR  Dn F ) ` 
0 )  =  A )
3227, 31feq12d 5710 . . . . 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 757 . . . . . . . . 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 4123 . . . . . . . . . . . . 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 756 . . . . . . . . . . . 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 22204 . . . . . . . . . . . 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 1229 . . . . . . . . . . 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 3525 . . . . . . . . . 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 755 . . . . . . . . . 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 3499 . . . . . . . . 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 22227 . . . . . . . . 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 22201 . . . . . . . . . 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 1229 . . . . . . . . 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 5195 . . . . . . . . 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 5710 . . . . . . . 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 10965 . . 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 1194 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 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   {cpr 4016   dom cdm 4989   -->wf 5574   ` cfv 5578  (class class class)co 6281    ^pm cpm 7423   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498   NN0cn0 10801    _D cdv 22140    Dncdvn 22141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-icc 11545  df-fz 11682  df-seq 12087  df-exp 12146  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-plusg 14587  df-mulr 14588  df-starv 14589  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-rest 14697  df-topn 14698  df-topgen 14718  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-xms 20696  df-ms 20697  df-cncf 21255  df-limc 22143  df-dv 22144  df-dvn 22145
This theorem is referenced by:  taylthlem2  22641
  Copyright terms: Public domain W3C validator