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Theorem taylf 19756
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 19754 . . . . . 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 447 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
98snssd 3776 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
101, 2, 3, 4, 5taylfvallem 19753 . . . . . . . . 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 4808 . . . . . . . . 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 642 . . . . . . . 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 2639 . . . . . . 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 3959 . . . . . . 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 203 . . . . . 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 3225 . . . . 5  |-  ( ph  ->  T  C_  ( CC  X.  CC ) )
17 relxp 4810 . . . . 5  |-  Rel  ( CC  X.  CC )
18 relss 4791 . . . . 5  |-  ( T 
C_  ( CC  X.  CC )  ->  ( Rel  ( CC  X.  CC )  ->  Rel  T )
)
1916, 17, 18ee10 1366 . . . 4  |-  ( ph  ->  Rel  T )
201, 2, 3, 4, 5, 6eltayl 19755 . . . . . . . 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 198 . . . . . . 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 1621 . . . . . 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 16399 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
24 cnrng 16412 . . . . . . . . . 10  |-fld  e.  Ring
25 rngcmn 15387 . . . . . . . . . 10  |-  (fld  e.  Ring  ->fld  e. CMnd )
2624, 25mp1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
27 cnfldtps 18303 . . . . . . . . . 10  |-fld  e.  TopSp
2827a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
29 ovex 5899 . . . . . . . . . . 11  |-  ( 0 [,] N )  e. 
_V
3029inex1 4171 . . . . . . . . . 10  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
3130a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V )
321, 2, 3, 4, 5taylfvallem1 19752 . . . . . . . . . 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 2296 . . . . . . . . . 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 5700 . . . . . . . . 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 2296 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3635cnfldhaus 18310 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Haus
3736a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( TopOpen ` fld )  e.  Haus )
3823, 26, 28, 31, 34, 35, 37haustsms 17834 . . . . . . . 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 423 . . . . . . 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 2214 . . . . . . 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 203 . . . . . 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 2202 . . . . . 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 56 . . . . 5  |-  ( ph  ->  E* y  x T y )
4443alrimiv 1621 . . . 4  |-  ( ph  ->  A. x E* y  x T y )
45 dffun6 5286 . . . 4  |-  ( Fun 
T  <->  ( Rel  T  /\  A. x E* y  x T y ) )
4619, 44, 45sylanbrc 645 . . 3  |-  ( ph  ->  Fun  T )
47 funfn 5299 . . 3  |-  ( Fun 
T  <->  T  Fn  dom  T )
4846, 47sylib 188 . 2  |-  ( ph  ->  T  Fn  dom  T
)
49 rnss 4923 . . . 4  |-  ( T 
C_  ( CC  X.  CC )  ->  ran  T  C_ 
ran  ( CC  X.  CC ) )
5016, 49syl 15 . . 3  |-  ( ph  ->  ran  T  C_  ran  ( CC  X.  CC ) )
51 rnxpss 5124 . . 3  |-  ran  ( CC  X.  CC )  C_  CC
5250, 51syl6ss 3204 . 2  |-  ( ph  ->  ran  T  C_  CC )
53 df-f 5275 . 2  |-  ( T : dom  T --> CC  <->  ( T  Fn  dom  T  /\  ran  T 
C_  CC ) )
5448, 52, 53sylanbrc 645 1  |-  ( ph  ->  T : dom  T --> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   E*wmo 2157   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   {cpr 3654   U_ciun 3921   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   dom cdm 4705   ran crn 4706   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    +oocpnf 8880    - cmin 9053    / cdiv 9439   NN0cn0 9981   ZZcz 10040   [,]cicc 10675   ^cexp 11120   !cfa 11304   TopOpenctopn 13342  CMndccmn 15105   Ringcrg 15353  ℂfldccnfld 16393   TopSpctps 16650   Hauscha 17052   tsums ctsu 17824    D ncdvn 19230   Tayl ctayl 19748
This theorem is referenced by:  tayl0  19757
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  ax-addf 8832  ax-mulf 8833
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-se 4369  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-isom 5280  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-oi 7241  df-card 7588  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-fzo 10887  df-seq 11063  df-exp 11121  df-fac 11305  df-hash 11354  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-sets 13170  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-0g 13420  df-gsum 13421  df-mnd 14383  df-grp 14505  df-minusg 14506  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  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-tsms 17825  df-xms 17901  df-ms 17902  df-limc 19232  df-dv 19233  df-dvn 19234  df-tayl 19750
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