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Theorem taylfvallem1 21844
Description: Lemma for taylfval 21846. (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 9384 . . . . . . . 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 7251 . . . . . . 7  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
84, 1, 5, 6, 7syl22anc 1219 . . . . . 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 3592 . . . . . . 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 3375 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
13 inss1 3591 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
1413, 11sseldi 3375 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
15 0xr 9451 . . . . . . . . 9  |-  0  e.  RR*
16 taylfval.n . . . . . . . . . . 11  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
17 nn0re 10609 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  RR )
1817rexrd 9454 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e. 
RR* )
19 id 22 . . . . . . . . . . . . 13  |-  ( N  = +oo  ->  N  = +oo )
20 pnfxr 11113 . . . . . . . . . . . . 13  |- +oo  e.  RR*
2119, 20syl6eqel 2531 . . . . . . . . . . . 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 11365 . . . . . . . . 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 1001 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
29 elnn0z 10680 . . . . . 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 21423 . . . . 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 1218 . . . 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 5865 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
36 faccl 12082 . . . . 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 10359 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
3937nnne0d 10387 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
4035, 38, 39divcld 10128 . 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 21424 . . . . . . . 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 1218 . . . . . . 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 5584 . . . . . . . 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 3413 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  Dn F ) `  k )  C_  A
)
48 recnprss 21401 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
491, 48syl 16 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
506, 49sstrd 3387 . . . . . . 7  |-  ( ph  ->  A  C_  CC )
5150ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  A  C_  CC )
5247, 51sstrd 3387 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  Dn F ) `  k )  C_  CC )
5352, 34sseldd 3378 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  CC )
5441, 53subcld 9740 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( X  -  B )  e.  CC )
5554, 30expcld 12029 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( X  -  B
) ^ k )  e.  CC )
5640, 55mulcld 9427 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2993    i^i cin 3348    C_ wss 3349   {cpr 3900   class class class wbr 4313   dom cdm 4861   -->wf 5435   ` cfv 5439  (class class class)co 6112    ^pm cpm 7236   CCcc 9301   RRcr 9302   0cc0 9303    x. cmul 9308   +oocpnf 9436   RR*cxr 9438    <_ cle 9440    - cmin 9616    / cdiv 10014   NNcn 10343   NN0cn0 10600   ZZcz 10667   [,]cicc 11324   ^cexp 11886   !cfa 12072    Dncdvn 21361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fi 7682  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-icc 11328  df-fz 11459  df-seq 11828  df-exp 11887  df-fac 12073  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-rest 14382  df-topn 14383  df-topgen 14403  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cnp 18854  df-haus 18941  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-xms 19917  df-ms 19918  df-limc 21363  df-dv 21364  df-dvn 21365
This theorem is referenced by:  taylfvallem  21845  taylf  21848  taylplem2  21851  taylpfval  21852
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