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Theorem dvn2bss 21304
Description: An N-times differentiable point is an M-times differentiable point, if  M  <_  N. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
dvn2bss  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  N )  C_  dom  ( ( S  Dn F ) `  M ) )

Proof of Theorem dvn2bss
StepHypRef Expression
1 simp1 983 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  S  e.  { RR ,  CC } )
2 simp2 984 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  F  e.  ( CC  ^pm  S
) )
3 elfznn0 11477 . . . . . 6  |-  ( M  e.  ( 0 ... N )  ->  M  e.  NN0 )
433ad2ant3 1006 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  M  e.  NN0 )
5 elfzuz3 11446 . . . . . . 7  |-  ( M  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  M )
)
653ad2ant3 1006 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  N  e.  ( ZZ>= `  M )
)
7 uznn0sub 10888 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  -  M )  e.  NN0 )
86, 7syl 16 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  ( N  -  M )  e.  NN0 )
9 dvnadd 21303 . . . . 5  |-  ( ( ( 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
) ) ) )
101, 2, 4, 8, 9syl22anc 1214 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  ( N  -  M
) )  =  ( ( S  Dn
F ) `  ( M  +  ( N  -  M ) ) ) )
114nn0cnd 10634 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  M  e.  CC )
12 elfzuz2 11452 . . . . . . . . 9  |-  ( M  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  0 )
)
13123ad2ant3 1006 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  N  e.  ( ZZ>= `  0 )
)
14 nn0uz 10891 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
1513, 14syl6eleqr 2532 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  N  e.  NN0 )
1615nn0cnd 10634 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  N  e.  CC )
1711, 16pncan3d 9718 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  ( M  +  ( N  -  M ) )  =  N )
1817fveq2d 5692 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  (
( S  Dn
F ) `  ( M  +  ( N  -  M ) ) )  =  ( ( S  Dn F ) `
 N ) )
1910, 18eqtrd 2473 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  ( N  -  M
) )  =  ( ( S  Dn
F ) `  N
) )
2019dmeqd 5038 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 ( N  -  M ) )  =  dom  ( ( S  Dn F ) `
 N ) )
21 cnex 9359 . . . . 5  |-  CC  e.  _V
2221a1i 11 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  CC  e.  _V )
23 dvnf 21301 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  NN0 )  ->  ( ( S  Dn F ) `
 M ) : dom  ( ( S  Dn F ) `
 M ) --> CC )
243, 23syl3an3 1248 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  (
( S  Dn
F ) `  M
) : dom  (
( S  Dn
F ) `  M
) --> CC )
25 dvnbss 21302 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  NN0 )  ->  dom  ( ( S  Dn F ) `  M ) 
C_  dom  F )
263, 25syl3an3 1248 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  M )  C_  dom  F )
27 elpmi 7227 . . . . . . 7  |-  ( F  e.  ( CC  ^pm  S )  ->  ( F : dom  F --> CC  /\  dom  F  C_  S )
)
28273ad2ant2 1005 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  S )
)
2928simprd 460 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  F 
C_  S )
3026, 29sstrd 3363 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  M )  C_  S
)
31 elpm2r 7226 . . . 4  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( ( ( S  Dn F ) `
 M ) : dom  ( ( S  Dn F ) `
 M ) --> CC 
/\  dom  ( ( S  Dn F ) `
 M )  C_  S ) )  -> 
( ( S  Dn F ) `  M )  e.  ( CC  ^pm  S )
)
3222, 1, 24, 30, 31syl22anc 1214 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  (
( S  Dn
F ) `  M
)  e.  ( CC 
^pm  S ) )
33 dvnbss 21302 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  (
( S  Dn
F ) `  M
)  e.  ( CC 
^pm  S )  /\  ( N  -  M
)  e.  NN0 )  ->  dom  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  ( N  -  M ) ) 
C_  dom  ( ( S  Dn F ) `
 M ) )
341, 32, 8, 33syl3anc 1213 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 ( N  -  M ) )  C_  dom  ( ( S  Dn F ) `  M ) )
3520, 34eqsstr3d 3388 1  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  N )  C_  dom  ( ( S  Dn F ) `  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   {cpr 3876   dom cdm 4836   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^pm cpm 7211   CCcc 9276   RRcr 9277   0cc0 9278    + caddc 9281    - cmin 9591   NN0cn0 10575   ZZ>=cuz 10857   ...cfz 11433    Dncdvn 21239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-icc 11303  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-rest 14357  df-topn 14358  df-topgen 14378  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cnp 18732  df-haus 18819  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-xms 19795  df-ms 19796  df-limc 21241  df-dv 21242  df-dvn 21243
This theorem is referenced by:  taylplem1  21771  taylply2  21776  taylply  21777  taylthlem1  21781  taylthlem2  21782
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