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Theorem taylfvallem1 20265
Description: Lemma for taylfval 20267. (Contributed by Mario Carneiro, 30-Dec-2016.)
Hypotheses
Ref Expression
taylfval.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
taylfval.f  |-  ( ph  ->  F : A --> CC )
taylfval.a  |-  ( ph  ->  A  C_  S )
taylfval.n  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
taylfval.b  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
Assertion
Ref Expression
taylfvallem1  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( X  -  B
) ^ k ) )  e.  CC )
Distinct variable groups:    B, k    k, F    ph, k    k, N    S, k    k, X
Allowed substitution hint:    A( k)

Proof of Theorem taylfvallem1
StepHypRef Expression
1 taylfval.s . . . . . 6  |-  ( ph  ->  S  e.  { RR ,  CC } )
21ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  S  e.  { RR ,  CC } )
3 cnex 9063 . . . . . . . 8  |-  CC  e.  _V
43a1i 11 . . . . . . 7  |-  ( ph  ->  CC  e.  _V )
5 taylfval.f . . . . . . 7  |-  ( ph  ->  F : A --> CC )
6 taylfval.a . . . . . . 7  |-  ( ph  ->  A  C_  S )
7 elpm2r 7026 . . . . . . 7  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
84, 1, 5, 6, 7syl22anc 1185 . . . . . 6  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
98ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F  e.  ( CC  ^pm  S
) )
10 inss2 3554 . . . . . . 7  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ZZ
11 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )
1210, 11sseldi 3338 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
13 inss1 3553 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
1413, 11sseldi 3338 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
15 0xr 9123 . . . . . . . . 9  |-  0  e.  RR*
16 taylfval.n . . . . . . . . . . 11  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
17 nn0re 10222 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  RR )
1817rexrd 9126 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e. 
RR* )
19 id 20 . . . . . . . . . . . . 13  |-  ( N  =  +oo  ->  N  =  +oo )
20 pnfxr 10705 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
2119, 20syl6eqel 2523 . . . . . . . . . . . 12  |-  ( N  =  +oo  ->  N  e.  RR* )
2218, 21jaoi 369 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  \/  N  =  +oo )  ->  N  e.  RR* )
2316, 22syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  RR* )
2423ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  N  e.  RR* )
25 elicc1 10952 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  N  e.  RR* )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
2615, 24, 25sylancr 645 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
2714, 26mpbid 202 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  RR*  /\  0  <_  k  /\  k  <_  N ) )
2827simp2d 970 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
29 elnn0z 10286 . . . . . 6  |-  ( k  e.  NN0  <->  ( k  e.  ZZ  /\  0  <_ 
k ) )
3012, 28, 29sylanbrc 646 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  NN0 )
31 dvnf 19805 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  ( ( S  D n F ) `
 k ) : dom  ( ( S  D n F ) `
 k ) --> CC )
322, 9, 30, 31syl3anc 1184 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  D n F ) `  k
) : dom  (
( S  D n F ) `  k
) --> CC )
33 taylfval.b . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
3433adantlr 696 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
3532, 34ffvelrnd 5863 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  D n F ) `  k
) `  B )  e.  CC )
36 faccl 11568 . . . . 5  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
3730, 36syl 16 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
3837nncnd 10008 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
3937nnne0d 10036 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
4035, 38, 39divcld 9782 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
41 simplr 732 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  X  e.  CC )
42 dvnbss 19806 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  dom  ( ( S  D n F ) `  k ) 
C_  dom  F )
432, 9, 30, 42syl3anc 1184 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  D n F ) `  k
)  C_  dom  F )
445ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F : A --> CC )
45 fdm 5587 . . . . . . . 8  |-  ( F : A --> CC  ->  dom 
F  =  A )
4644, 45syl 16 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  F  =  A )
4743, 46sseqtrd 3376 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  D n F ) `  k
)  C_  A )
48 recnprss 19783 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
491, 48syl 16 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
506, 49sstrd 3350 . . . . . . 7  |-  ( ph  ->  A  C_  CC )
5150ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  A  C_  CC )
5247, 51sstrd 3350 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  D n F ) `  k
)  C_  CC )
5352, 34sseldd 3341 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  CC )
5441, 53subcld 9403 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( X  -  B )  e.  CC )
5554, 30expcld 11515 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( X  -  B
) ^ k )  e.  CC )
5640, 55mulcld 9100 1  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( X  -  B
) ^ k ) )  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   {cpr 3807   class class class wbr 4204   dom cdm 4870   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    +oocpnf 9109   RR*cxr 9111    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   [,]cicc 10911   ^cexp 11374   !cfa 11558    D ncdvn 19743
This theorem is referenced by:  taylfvallem  20266  taylf  20269  taylplem2  20272  taylpfval  20273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-icc 10915  df-fz 11036  df-seq 11316  df-exp 11375  df-fac 11559  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-rest 13642  df-topn 13643  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cnp 17284  df-haus 17371  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-limc 19745  df-dv 19746  df-dvn 19747
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