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Theorem taylfvallem1 22617
Description: Lemma for taylfval 22619. (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  Dn F ) `
 k ) )
Assertion
Ref Expression
taylfvallem1  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn 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 725 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  S  e.  { RR ,  CC } )
3 cnex 9585 . . . . . . . 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 7448 . . . . . . 7  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
84, 1, 5, 6, 7syl22anc 1229 . . . . . 6  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
98ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F  e.  ( CC  ^pm  S
) )
10 inss2 3724 . . . . . . 7  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ZZ
11 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )
1210, 11sseldi 3507 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
13 inss1 3723 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
1413, 11sseldi 3507 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
15 0xr 9652 . . . . . . . . 9  |-  0  e.  RR*
16 taylfval.n . . . . . . . . . . 11  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
17 nn0re 10816 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  RR )
1817rexrd 9655 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e. 
RR* )
19 id 22 . . . . . . . . . . . . 13  |-  ( N  = +oo  ->  N  = +oo )
20 pnfxr 11333 . . . . . . . . . . . . 13  |- +oo  e.  RR*
2119, 20syl6eqel 2563 . . . . . . . . . . . 12  |-  ( N  = +oo  ->  N  e.  RR* )
2218, 21jaoi 379 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  N  e.  RR* )
2316, 22syl 16 . . . . . . . . . 10  |-  ( ph  ->  N  e.  RR* )
2423ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  N  e.  RR* )
25 elicc1 11585 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  N  e.  RR* )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
2615, 24, 25sylancr 663 . . . . . . . 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 210 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  RR*  /\  0  <_  k  /\  k  <_  N ) )
2827simp2d 1009 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
29 elnn0z 10889 . . . . . 6  |-  ( k  e.  NN0  <->  ( k  e.  ZZ  /\  0  <_ 
k ) )
3012, 28, 29sylanbrc 664 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  NN0 )
31 dvnf 22196 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  ( ( S  Dn F ) `
 k ) : dom  ( ( S  Dn F ) `
 k ) --> CC )
322, 9, 30, 31syl3anc 1228 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  Dn
F ) `  k
) : dom  (
( S  Dn
F ) `  k
) --> CC )
33 taylfval.b . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  Dn F ) `
 k ) )
3433adantlr 714 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  Dn F ) `
 k ) )
3532, 34ffvelrnd 6033 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
36 faccl 12343 . . . . 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 10564 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
3937nnne0d 10592 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
4035, 38, 39divcld 10332 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
41 simplr 754 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  X  e.  CC )
42 dvnbss 22197 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  dom  ( ( S  Dn F ) `  k ) 
C_  dom  F )
432, 9, 30, 42syl3anc 1228 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  Dn F ) `  k )  C_  dom  F )
445ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F : A --> CC )
45 fdm 5741 . . . . . . . 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 3545 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  Dn F ) `  k )  C_  A
)
48 recnprss 22174 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
491, 48syl 16 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
506, 49sstrd 3519 . . . . . . 7  |-  ( ph  ->  A  C_  CC )
5150ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  A  C_  CC )
5247, 51sstrd 3519 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  Dn F ) `  k )  C_  CC )
5352, 34sseldd 3510 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  CC )
5441, 53subcld 9942 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( X  -  B )  e.  CC )
5554, 30expcld 12290 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( X  -  B
) ^ k )  e.  CC )
5640, 55mulcld 9628 1  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( X  -  B
) ^ k ) )  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481   {cpr 4035   class class class wbr 4453   dom cdm 5005   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^pm cpm 7433   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509   +oocpnf 9637   RR*cxr 9639    <_ cle 9641    - cmin 9817    / cdiv 10218   NNcn 10548   NN0cn0 10807   ZZcz 10876   [,]cicc 11544   ^cexp 12146   !cfa 12333    Dncdvn 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-icc 11548  df-fz 11685  df-seq 12088  df-exp 12147  df-fac 12334  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-plusg 14584  df-mulr 14585  df-starv 14586  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-rest 14694  df-topn 14695  df-topgen 14715  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cnp 19595  df-haus 19682  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-limc 22136  df-dv 22137  df-dvn 22138
This theorem is referenced by:  taylfvallem  22618  taylf  22621  taylplem2  22624  taylpfval  22625
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