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Theorem taylfvallem1 22877
Description: Lemma for taylfval 22879. (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 9590 . . . . . . . 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 7455 . . . . . . 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 3715 . . . . . . 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 3497 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
13 inss1 3714 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
1413, 11sseldi 3497 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
15 0xr 9657 . . . . . . . . 9  |-  0  e.  RR*
16 taylfval.n . . . . . . . . . . 11  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
17 nn0re 10825 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  RR )
1817rexrd 9660 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e. 
RR* )
19 id 22 . . . . . . . . . . . . 13  |-  ( N  = +oo  ->  N  = +oo )
20 pnfxr 11346 . . . . . . . . . . . . 13  |- +oo  e.  RR*
2119, 20syl6eqel 2553 . . . . . . . . . . . 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 11598 . . . . . . . . 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 10898 . . . . . 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 22455 . . . . 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 12365 . . . . 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 10572 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
3937nnne0d 10601 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
4035, 38, 39divcld 10341 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
41 simplr 755 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  X  e.  CC )
42 dvnbss 22456 . . . . . . . 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 3535 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  Dn F ) `  k )  C_  A
)
48 recnprss 22433 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
491, 48syl 16 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
506, 49sstrd 3509 . . . . . . 7  |-  ( ph  ->  A  C_  CC )
5150ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  A  C_  CC )
5247, 51sstrd 3509 . . . . 5  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  dom  ( ( S  Dn F ) `  k )  C_  CC )
5352, 34sseldd 3500 . . . 4  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  CC )
5441, 53subcld 9950 . . 3  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( X  -  B )  e.  CC )
5554, 30expcld 12312 . 2  |-  ( ( ( ph  /\  X  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( X  -  B
) ^ k )  e.  CC )
5640, 55mulcld 9633 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 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    C_ wss 3471   {cpr 4034   class class class wbr 4456   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^pm cpm 7439   CCcc 9507   RRcr 9508   0cc0 9509    x. cmul 9514   +oocpnf 9642   RR*cxr 9644    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   NN0cn0 10816   ZZcz 10885   [,]cicc 11557   ^cexp 12168   !cfa 12355    Dncdvn 22393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-icc 11561  df-fz 11698  df-seq 12110  df-exp 12169  df-fac 12356  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-plusg 14724  df-mulr 14725  df-starv 14726  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-rest 14839  df-topn 14840  df-topgen 14860  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cnp 19855  df-haus 19942  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-limc 22395  df-dv 22396  df-dvn 22397
This theorem is referenced by:  taylfvallem  22878  taylf  22881  taylplem2  22884  taylpfval  22885
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