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Theorem taylf 20138
Description: The Taylor series defines a function on a subset of the complexes. (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 ) )
taylfval.t  |-  T  =  ( N ( S Tayl 
F ) B )
Assertion
Ref Expression
taylf  |-  ( ph  ->  T : dom  T --> CC )
Distinct variable groups:    B, k    k, F    ph, k    k, N    S, k
Allowed substitution hints:    A( k)    T( k)

Proof of Theorem taylf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 taylfval.s . . . . . . 7  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 taylfval.f . . . . . . 7  |-  ( ph  ->  F : A --> CC )
3 taylfval.a . . . . . . 7  |-  ( ph  ->  A  C_  S )
4 taylfval.n . . . . . . 7  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
5 taylfval.b . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
6 taylfval.t . . . . . . 7  |-  T  =  ( N ( S Tayl 
F ) B )
71, 2, 3, 4, 5, 6taylfval 20136 . . . . . 6  |-  ( ph  ->  T  =  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
8 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
98snssd 3880 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
101, 2, 3, 4, 5taylfvallem 20135 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) 
C_  CC )
11 xpss12 4915 . . . . . . . . 9  |-  ( ( { x }  C_  CC  /\  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) 
C_  CC )  -> 
( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )  C_  ( CC  X.  CC ) )
129, 10, 11syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) )  C_  ( CC  X.  CC ) )
1312ralrimiva 2726 . . . . . . 7  |-  ( ph  ->  A. x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC ) )
14 iunss 4067 . . . . . . 7  |-  ( U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC )  <->  A. x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC ) )
1513, 14sylibr 204 . . . . . 6  |-  ( ph  ->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC ) )
167, 15eqsstrd 3319 . . . . 5  |-  ( ph  ->  T  C_  ( CC  X.  CC ) )
17 relxp 4917 . . . . 5  |-  Rel  ( CC  X.  CC )
18 relss 4897 . . . . 5  |-  ( T 
C_  ( CC  X.  CC )  ->  ( Rel  ( CC  X.  CC )  ->  Rel  T )
)
1916, 17, 18ee10 1382 . . . 4  |-  ( ph  ->  Rel  T )
201, 2, 3, 4, 5, 6eltayl 20137 . . . . . . . 8  |-  ( ph  ->  ( x T y  <-> 
( x  e.  CC  /\  y  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) ) )
2120biimpd 199 . . . . . . 7  |-  ( ph  ->  ( x T y  ->  ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) ) )
2221alrimiv 1638 . . . . . 6  |-  ( ph  ->  A. y ( x T y  ->  (
x  e.  CC  /\  y  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) ) )
23 cnfldbas 16624 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
24 cnrng 16640 . . . . . . . . . 10  |-fld  e.  Ring
25 rngcmn 15615 . . . . . . . . . 10  |-  (fld  e.  Ring  ->fld  e. CMnd )
2624, 25mp1i 12 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
27 cnfldtps 18677 . . . . . . . . . 10  |-fld  e.  TopSp
2827a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
29 ovex 6039 . . . . . . . . . . 11  |-  ( 0 [,] N )  e. 
_V
3029inex1 4279 . . . . . . . . . 10  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
3130a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V )
321, 2, 3, 4, 5taylfvallem1 20134 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) )  e.  CC )
33 eqid 2381 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )  =  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )
3432, 33fmptd 5826 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) : ( ( 0 [,] N )  i^i  ZZ ) --> CC )
35 eqid 2381 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3635cnfldhaus 18684 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Haus
3736a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( TopOpen ` fld )  e.  Haus )
3823, 26, 28, 31, 34, 35, 37haustsms 18080 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  E* y 
y  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )
3938ex 424 . . . . . . 7  |-  ( ph  ->  ( x  e.  CC  ->  E* y  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
40 moanimv 2290 . . . . . . 7  |-  ( E* y ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )  <->  ( x  e.  CC  ->  E* y 
y  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
4139, 40sylibr 204 . . . . . 6  |-  ( ph  ->  E* y ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
42 moim 2278 . . . . . 6  |-  ( A. y ( x T y  ->  ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )  ->  ( E* y ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )  ->  E* y  x T y ) )
4322, 41, 42sylc 58 . . . . 5  |-  ( ph  ->  E* y  x T y )
4443alrimiv 1638 . . . 4  |-  ( ph  ->  A. x E* y  x T y )
45 dffun6 5403 . . . 4  |-  ( Fun 
T  <->  ( Rel  T  /\  A. x E* y  x T y ) )
4619, 44, 45sylanbrc 646 . . 3  |-  ( ph  ->  Fun  T )
47 funfn 5416 . . 3  |-  ( Fun 
T  <->  T  Fn  dom  T )
4846, 47sylib 189 . 2  |-  ( ph  ->  T  Fn  dom  T
)
49 rnss 5032 . . . 4  |-  ( T 
C_  ( CC  X.  CC )  ->  ran  T  C_ 
ran  ( CC  X.  CC ) )
5016, 49syl 16 . . 3  |-  ( ph  ->  ran  T  C_  ran  ( CC  X.  CC ) )
51 rnxpss 5235 . . 3  |-  ran  ( CC  X.  CC )  C_  CC
5250, 51syl6ss 3297 . 2  |-  ( ph  ->  ran  T  C_  CC )
53 df-f 5392 . 2  |-  ( T : dom  T --> CC  <->  ( T  Fn  dom  T  /\  ran  T 
C_  CC ) )
5448, 52, 53sylanbrc 646 1  |-  ( ph  ->  T : dom  T --> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   E*wmo 2233   A.wral 2643   _Vcvv 2893    i^i cin 3256    C_ wss 3257   {csn 3751   {cpr 3752   U_ciun 4029   class class class wbr 4147    e. cmpt 4201    X. cxp 4810   dom cdm 4812   ran crn 4813   Rel wrel 4817   Fun wfun 5382    Fn wfn 5383   -->wf 5384   ` cfv 5388  (class class class)co 6014   CCcc 8915   RRcr 8916   0cc0 8917    x. cmul 8922    +oocpnf 9044    - cmin 9217    / cdiv 9603   NN0cn0 10147   ZZcz 10208   [,]cicc 10845   ^cexp 11303   !cfa 11487   TopOpenctopn 13570  CMndccmn 15333   Ringcrg 15581  ℂfldccnfld 16620   TopSpctps 16878   Hauscha 17288   tsums ctsu 18070    D ncdvn 19612   Tayl ctayl 20130
This theorem is referenced by:  tayl0  20139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-inf2 7523  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994  ax-pre-sup 8995  ax-addf 8996  ax-mulf 8997
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-iin 4032  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-se 4477  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-isom 5397  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-er 6835  df-map 6950  df-pm 6951  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-fi 7345  df-sup 7375  df-oi 7406  df-card 7753  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-div 9604  df-nn 9927  df-2 9984  df-3 9985  df-4 9986  df-5 9987  df-6 9988  df-7 9989  df-8 9990  df-9 9991  df-10 9992  df-n0 10148  df-z 10209  df-dec 10309  df-uz 10415  df-q 10501  df-rp 10539  df-xneg 10636  df-xadd 10637  df-xmul 10638  df-icc 10849  df-fz 10970  df-fzo 11060  df-seq 11245  df-exp 11304  df-fac 11488  df-hash 11540  df-cj 11825  df-re 11826  df-im 11827  df-sqr 11961  df-abs 11962  df-struct 13392  df-ndx 13393  df-slot 13394  df-base 13395  df-sets 13396  df-plusg 13463  df-mulr 13464  df-starv 13465  df-tset 13469  df-ple 13470  df-ds 13472  df-unif 13473  df-rest 13571  df-topn 13572  df-topgen 13588  df-0g 13648  df-gsum 13649  df-mnd 14611  df-grp 14733  df-minusg 14734  df-cntz 15037  df-cmn 15335  df-abl 15336  df-mgp 15570  df-rng 15584  df-cring 15585  df-ur 15586  df-xmet 16613  df-met 16614  df-bl 16615  df-mopn 16616  df-fbas 16617  df-fg 16618  df-cnfld 16621  df-top 16880  df-bases 16882  df-topon 16883  df-topsp 16884  df-cld 17000  df-ntr 17001  df-cls 17002  df-nei 17079  df-lp 17117  df-perf 17118  df-cnp 17208  df-haus 17295  df-fil 17793  df-fm 17885  df-flim 17886  df-flf 17887  df-tsms 18071  df-xms 18253  df-ms 18254  df-limc 19614  df-dv 19615  df-dvn 19616  df-tayl 20132
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