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Theorem dvntaylp0 22951
Description: The first  N derivatives of the Taylor polynomial at  B match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.)
Hypotheses
Ref Expression
dvntaylp0.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
dvntaylp0.f  |-  ( ph  ->  F : A --> CC )
dvntaylp0.a  |-  ( ph  ->  A  C_  S )
dvntaylp0.m  |-  ( ph  ->  M  e.  ( 0 ... N ) )
dvntaylp0.b  |-  ( ph  ->  B  e.  dom  (
( S  Dn
F ) `  N
) )
dvntaylp0.t  |-  T  =  ( N ( S Tayl 
F ) B )
Assertion
Ref Expression
dvntaylp0  |-  ( ph  ->  ( ( ( CC  Dn T ) `
 M ) `  B )  =  ( ( ( S  Dn F ) `  M ) `  B
) )

Proof of Theorem dvntaylp0
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 dvntaylp0.m . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ( 0 ... N ) )
2 elfz3nn0 11744 . . . . . . . . . . 11  |-  ( M  e.  ( 0 ... N )  ->  N  e.  NN0 )
31, 2syl 17 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
43nn0cnd 10815 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
5 elfznn0 11743 . . . . . . . . . . 11  |-  ( M  e.  ( 0 ... N )  ->  M  e.  NN0 )
61, 5syl 17 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
76nn0cnd 10815 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
84, 7npcand 9891 . . . . . . . 8  |-  ( ph  ->  ( ( N  -  M )  +  M
)  =  N )
98oveq1d 6249 . . . . . . 7  |-  ( ph  ->  ( ( ( N  -  M )  +  M ) ( S Tayl 
F ) B )  =  ( N ( S Tayl  F ) B ) )
10 dvntaylp0.t . . . . . . 7  |-  T  =  ( N ( S Tayl 
F ) B )
119, 10syl6eqr 2461 . . . . . 6  |-  ( ph  ->  ( ( ( N  -  M )  +  M ) ( S Tayl 
F ) B )  =  T )
1211oveq2d 6250 . . . . 5  |-  ( ph  ->  ( CC  Dn
( ( ( N  -  M )  +  M ) ( S Tayl 
F ) B ) )  =  ( CC  Dn T ) )
1312fveq1d 5807 . . . 4  |-  ( ph  ->  ( ( CC  Dn ( ( ( N  -  M )  +  M ) ( S Tayl  F ) B ) ) `  M
)  =  ( ( CC  Dn T ) `  M ) )
14 dvntaylp0.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
15 dvntaylp0.f . . . . 5  |-  ( ph  ->  F : A --> CC )
16 dvntaylp0.a . . . . 5  |-  ( ph  ->  A  C_  S )
17 fznn0sub 11688 . . . . . 6  |-  ( M  e.  ( 0 ... N )  ->  ( N  -  M )  e.  NN0 )
181, 17syl 17 . . . . 5  |-  ( ph  ->  ( N  -  M
)  e.  NN0 )
19 dvntaylp0.b . . . . . 6  |-  ( ph  ->  B  e.  dom  (
( S  Dn
F ) `  N
) )
208fveq2d 5809 . . . . . . 7  |-  ( ph  ->  ( ( S  Dn F ) `  ( ( N  -  M )  +  M
) )  =  ( ( S  Dn
F ) `  N
) )
2120dmeqd 5147 . . . . . 6  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 ( ( N  -  M )  +  M ) )  =  dom  ( ( S  Dn F ) `
 N ) )
2219, 21eleqtrrd 2493 . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  Dn
F ) `  (
( N  -  M
)  +  M ) ) )
2314, 15, 16, 6, 18, 22dvntaylp 22950 . . . 4  |-  ( ph  ->  ( ( CC  Dn ( ( ( N  -  M )  +  M ) ( S Tayl  F ) B ) ) `  M
)  =  ( ( N  -  M ) ( S Tayl  ( ( S  Dn F ) `  M ) ) B ) )
2413, 23eqtr3d 2445 . . 3  |-  ( ph  ->  ( ( CC  Dn T ) `  M )  =  ( ( N  -  M
) ( S Tayl  (
( S  Dn
F ) `  M
) ) B ) )
2524fveq1d 5807 . 2  |-  ( ph  ->  ( ( ( CC  Dn T ) `
 M ) `  B )  =  ( ( ( N  -  M ) ( S Tayl  ( ( S  Dn F ) `  M ) ) B ) `  B ) )
26 cnex 9523 . . . . . . 7  |-  CC  e.  _V
2726a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
28 elpm2r 7394 . . . . . 6  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
2927, 14, 15, 16, 28syl22anc 1231 . . . . 5  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
30 dvnf 22514 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  NN0 )  ->  ( ( S  Dn F ) `
 M ) : dom  ( ( S  Dn F ) `
 M ) --> CC )
3114, 29, 6, 30syl3anc 1230 . . . 4  |-  ( ph  ->  ( ( S  Dn F ) `  M ) : dom  ( ( S  Dn F ) `  M ) --> CC )
32 dvnbss 22515 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  NN0 )  ->  dom  ( ( S  Dn F ) `  M ) 
C_  dom  F )
3314, 29, 6, 32syl3anc 1230 . . . . . 6  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 M )  C_  dom  F )
34 fdm 5674 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
3515, 34syl 17 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
3633, 35sseqtrd 3477 . . . . 5  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 M )  C_  A )
3736, 16sstrd 3451 . . . 4  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 M )  C_  S )
3818orcd 390 . . . 4  |-  ( ph  ->  ( ( N  -  M )  e.  NN0  \/  ( N  -  M
)  = +oo )
)
39 dvnadd 22516 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( M  e.  NN0  /\  ( N  -  M
)  e.  NN0 )
)  ->  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  ( N  -  M ) )  =  ( ( S  Dn F ) `
 ( M  +  ( N  -  M
) ) ) )
4014, 29, 6, 18, 39syl22anc 1231 . . . . . . . 8  |-  ( ph  ->  ( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 ( N  -  M ) )  =  ( ( S  Dn F ) `  ( M  +  ( N  -  M )
) ) )
417, 4pncan3d 9890 . . . . . . . . 9  |-  ( ph  ->  ( M  +  ( N  -  M ) )  =  N )
4241fveq2d 5809 . . . . . . . 8  |-  ( ph  ->  ( ( S  Dn F ) `  ( M  +  ( N  -  M )
) )  =  ( ( S  Dn
F ) `  N
) )
4340, 42eqtrd 2443 . . . . . . 7  |-  ( ph  ->  ( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 ( N  -  M ) )  =  ( ( S  Dn F ) `  N ) )
4443dmeqd 5147 . . . . . 6  |-  ( ph  ->  dom  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  ( N  -  M ) )  =  dom  ( ( S  Dn F ) `  N ) )
4519, 44eleqtrrd 2493 . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  ( N  -  M
) ) )
4614, 31, 37, 18, 45taylplem1 22942 . . . 4  |-  ( (
ph  /\  k  e.  ( ( 0 [,] ( N  -  M
) )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  k ) )
47 eqid 2402 . . . 4  |-  ( ( N  -  M ) ( S Tayl  ( ( S  Dn F ) `  M ) ) B )  =  ( ( N  -  M ) ( S Tayl  ( ( S  Dn F ) `  M ) ) B )
4814, 31, 37, 38, 46, 47tayl0 22941 . . 3  |-  ( ph  ->  ( B  e.  dom  ( ( N  -  M ) ( S Tayl  ( ( S  Dn F ) `  M ) ) B )  /\  ( ( ( N  -  M
) ( S Tayl  (
( S  Dn
F ) `  M
) ) B ) `
 B )  =  ( ( ( S  Dn F ) `
 M ) `  B ) ) )
4948simprd 461 . 2  |-  ( ph  ->  ( ( ( N  -  M ) ( S Tayl  ( ( S  Dn F ) `
 M ) ) B ) `  B
)  =  ( ( ( S  Dn
F ) `  M
) `  B )
)
5025, 49eqtrd 2443 1  |-  ( ph  ->  ( ( ( CC  Dn T ) `
 M ) `  B )  =  ( ( ( S  Dn F ) `  M ) `  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3058    C_ wss 3413   {cpr 3973   dom cdm 4942   -->wf 5521   ` cfv 5525  (class class class)co 6234    ^pm cpm 7378   CCcc 9440   RRcr 9441   0cc0 9442    + caddc 9445   +oocpnf 9575    - cmin 9761   NN0cn0 10756   ...cfz 11643    Dncdvn 22452   Tayl ctayl 22932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520  ax-addf 9521  ax-mulf 9522
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-of 6477  df-om 6639  df-1st 6738  df-2nd 6739  df-supp 6857  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-ixp 7428  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-fsupp 7784  df-fi 7825  df-sup 7855  df-oi 7889  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-9 10562  df-10 10563  df-n0 10757  df-z 10826  df-dec 10940  df-uz 11046  df-q 11146  df-rp 11184  df-xneg 11289  df-xadd 11290  df-xmul 11291  df-icc 11507  df-fz 11644  df-fzo 11768  df-seq 12062  df-exp 12121  df-fac 12308  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-sum 13565  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-mulr 14815  df-starv 14816  df-sca 14817  df-vsca 14818  df-ip 14819  df-tset 14820  df-ple 14821  df-ds 14823  df-unif 14824  df-hom 14825  df-cco 14826  df-rest 14929  df-topn 14930  df-0g 14948  df-gsum 14949  df-topgen 14950  df-pt 14951  df-prds 14954  df-xrs 15008  df-qtop 15013  df-imas 15014  df-xps 15016  df-mre 15092  df-mrc 15093  df-acs 15095  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-submnd 16183  df-grp 16273  df-minusg 16274  df-mulg 16276  df-cntz 16571  df-cmn 17016  df-abl 17017  df-mgp 17354  df-ur 17366  df-ring 17412  df-cring 17413  df-psmet 18623  df-xmet 18624  df-met 18625  df-bl 18626  df-mopn 18627  df-fbas 18628  df-fg 18629  df-cnfld 18633  df-top 19583  df-bases 19585  df-topon 19586  df-topsp 19587  df-cld 19704  df-ntr 19705  df-cls 19706  df-nei 19784  df-lp 19822  df-perf 19823  df-cn 19913  df-cnp 19914  df-haus 20001  df-tx 20247  df-hmeo 20440  df-fil 20531  df-fm 20623  df-flim 20624  df-flf 20625  df-tsms 20809  df-xms 21007  df-ms 21008  df-tms 21009  df-cncf 21566  df-limc 22454  df-dv 22455  df-dvn 22456  df-tayl 22934
This theorem is referenced by:  taylthlem1  22952  taylthlem2  22953
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