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Theorem taylfvallem1 19752
Description: Lemma for taylfval 19754. (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 706 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  S  e.  { RR ,  CC } )
3 cnex 8834 . . . . . . . 8  |-  CC  e.  _V
43a1i 10 . . . . . . 7  |-  ( ph  ->  CC  e.  _V )
5 taylfval.f . . . . . . 7  |-  ( ph  ->  F : A --> CC )
6 taylfval.a . . . . . . 7  |-  ( ph  ->  A  C_  S )
7 elpm2r 6804 . . . . . . 7  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
84, 1, 5, 6, 7syl22anc 1183 . . . . . 6  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
98ad2antrr 706 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F  e.  ( CC  ^pm  S
) )
10 inss2 3403 . . . . . . 7  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ZZ
11 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )
1210, 11sseldi 3191 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
13 inss1 3402 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
1413, 11sseldi 3191 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
15 0xr 8894 . . . . . . . . 9  |-  0  e.  RR*
16 taylfval.n . . . . . . . . . . 11  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
17 nn0re 9990 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  RR )
1817rexrd 8897 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e. 
RR* )
19 id 19 . . . . . . . . . . . . 13  |-  ( N  =  +oo  ->  N  =  +oo )
20 pnfxr 10471 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
2119, 20syl6eqel 2384 . . . . . . . . . . . 12  |-  ( N  =  +oo  ->  N  e.  RR* )
2218, 21jaoi 368 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  \/  N  =  +oo )  ->  N  e.  RR* )
2316, 22syl 15 . . . . . . . . . 10  |-  ( ph  ->  N  e.  RR* )
2423ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  N  e.  RR* )
25 elicc1 10716 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  N  e.  RR* )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
2615, 24, 25sylancr 644 . . . . . . . 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 201 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  RR*  /\  0  <_  k  /\  k  <_  N ) )
2827simp2d 968 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
29 elnn0z 10052 . . . . . 6  |-  ( k  e.  NN0  <->  ( k  e.  ZZ  /\  0  <_ 
k ) )
3012, 28, 29sylanbrc 645 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  NN0 )
31 dvnf 19292 . . . . 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 1182 . . . 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 695 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
3532, 34ffvelrnd 5682 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  D n F ) `  k
) `  B )  e.  CC )
36 faccl 11314 . . . . 5  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
3730, 36syl 15 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
3837nncnd 9778 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
3937nnne0d 9806 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
4035, 38, 39divcld 9552 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
41 simplr 731 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  X  e.  CC )
42 dvnbss 19293 . . . . . . . 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 1182 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  D n F ) `  k
)  C_  dom  F )
445ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F : A --> CC )
45 fdm 5409 . . . . . . . 8  |-  ( F : A --> CC  ->  dom 
F  =  A )
4644, 45syl 15 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  F  =  A )
4743, 46sseqtrd 3227 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  D n F ) `  k
)  C_  A )
48 recnprss 19270 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
491, 48syl 15 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
506, 49sstrd 3202 . . . . . . 7  |-  ( ph  ->  A  C_  CC )
5150ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  A  C_  CC )
5247, 51sstrd 3202 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  D n F ) `  k
)  C_  CC )
5352, 34sseldd 3194 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  CC )
5441, 53subcld 9173 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( X  -  B )  e.  CC )
5554, 30expcld 11261 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( X  -  B
) ^ k )  e.  CC )
5640, 55mulcld 8871 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {cpr 3654   class class class wbr 4039   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   [,]cicc 10675   ^cexp 11120   !cfa 11304    D ncdvn 19230
This theorem is referenced by:  taylfvallem  19753  taylf  19756  taylplem2  19759  taylpfval  19760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-fac 11305  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cnp 16974  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-limc 19232  df-dv 19233  df-dvn 19234
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