MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvn2bss Structured version   Unicode version

Theorem dvn2bss 22061
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 991 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  S  e.  { RR ,  CC } )
2 simp2 992 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  F  e.  ( CC  ^pm  S
) )
3 elfznn0 11759 . . . . . 6  |-  ( M  e.  ( 0 ... N )  ->  M  e.  NN0 )
433ad2ant3 1014 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  M  e.  NN0 )
5 elfzuz3 11674 . . . . . . 7  |-  ( M  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  M )
)
653ad2ant3 1014 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  N  e.  ( ZZ>= `  M )
)
7 uznn0sub 11102 . . . . . 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 22060 . . . . 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 1224 . . . 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 10843 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  M  e.  CC )
12 elfzuz2 11680 . . . . . . . . 9  |-  ( M  e.  ( 0 ... N )  ->  N  e.  ( ZZ>= `  0 )
)
13123ad2ant3 1014 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  N  e.  ( ZZ>= `  0 )
)
14 nn0uz 11105 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
1513, 14syl6eleqr 2559 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  N  e.  NN0 )
1615nn0cnd 10843 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  N  e.  CC )
1711, 16pncan3d 9922 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  ( M  +  ( N  -  M ) )  =  N )
1817fveq2d 5861 . . . 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 2501 . . 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 5196 . 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 9562 . . . . 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 22058 . . . . 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 1258 . . . 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 22059 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  NN0 )  ->  dom  ( ( S  Dn F ) `  M ) 
C_  dom  F )
263, 25syl3an3 1258 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  M )  C_  dom  F )
27 elpmi 7427 . . . . . . 7  |-  ( F  e.  ( CC  ^pm  S )  ->  ( F : dom  F --> CC  /\  dom  F  C_  S )
)
28273ad2ant2 1013 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  S )
)
2928simprd 463 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  F 
C_  S )
3026, 29sstrd 3507 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  M )  C_  S
)
31 elpm2r 7426 . . . 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 1224 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  M  e.  ( 0 ... N
) )  ->  (
( S  Dn
F ) `  M
)  e.  ( CC 
^pm  S ) )
33 dvnbss 22059 . . 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 1223 . 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 3532 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   {cpr 4022   dom cdm 4992   -->wf 5575   ` cfv 5579  (class class class)co 6275    ^pm cpm 7411   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    - cmin 9794   NN0cn0 10784   ZZ>=cuz 11071   ...cfz 11661    Dncdvn 21996
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fi 7860  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-icc 11525  df-fz 11662  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-starv 14559  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-rest 14667  df-topn 14668  df-topgen 14688  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cnp 19488  df-haus 19575  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-limc 21998  df-dv 21999  df-dvn 22000
This theorem is referenced by:  taylplem1  22485  taylply2  22490  taylply  22491  taylthlem1  22495  taylthlem2  22496
  Copyright terms: Public domain W3C validator