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Theorem taylf 23395
Description: The Taylor series defines a function on a subset of the complex numbers. (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 ) )
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  Dn F ) `
 k ) )
6 taylfval.t . . . . . . 7  |-  T  =  ( N ( S Tayl 
F ) B )
71, 2, 3, 4, 5, 6taylfval 23393 . . . . . 6  |-  ( ph  ->  T  =  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
8 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
98snssd 4108 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
101, 2, 3, 4, 5taylfvallem 23392 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) )  C_  CC )
11 xpss12 4945 . . . . . . . . 9  |-  ( ( { x }  C_  CC  /\  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) 
C_  CC )  -> 
( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )  C_  ( CC  X.  CC ) )
129, 10, 11syl2anc 673 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) )  C_  ( CC  X.  CC ) )
1312ralrimiva 2809 . . . . . . 7  |-  ( ph  ->  A. x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC ) )
14 iunss 4310 . . . . . . 7  |-  ( U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn 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  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC ) )
1513, 14sylibr 217 . . . . . 6  |-  ( ph  ->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC ) )
167, 15eqsstrd 3452 . . . . 5  |-  ( ph  ->  T  C_  ( CC  X.  CC ) )
17 relxp 4947 . . . . 5  |-  Rel  ( CC  X.  CC )
18 relss 4927 . . . . 5  |-  ( T 
C_  ( CC  X.  CC )  ->  ( Rel  ( CC  X.  CC )  ->  Rel  T )
)
1916, 17, 18mpisyl 21 . . . 4  |-  ( ph  ->  Rel  T )
201, 2, 3, 4, 5, 6eltayl 23394 . . . . . . . 8  |-  ( ph  ->  ( x T y  <-> 
( x  e.  CC  /\  y  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) ) ) ) )
2120biimpd 212 . . . . . . 7  |-  ( ph  ->  ( x T y  ->  ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) ) )
2221alrimiv 1781 . . . . . 6  |-  ( ph  ->  A. y ( x T y  ->  (
x  e.  CC  /\  y  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) ) ) ) )
23 cnfldbas 19051 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
24 cnring 19067 . . . . . . . . . 10  |-fld  e.  Ring
25 ringcmn 17889 . . . . . . . . . 10  |-  (fld  e.  Ring  ->fld  e. CMnd )
2624, 25mp1i 13 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
27 cnfldtps 21876 . . . . . . . . . 10  |-fld  e.  TopSp
2827a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
29 ovex 6336 . . . . . . . . . . 11  |-  ( 0 [,] N )  e. 
_V
3029inex1 4537 . . . . . . . . . 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 23391 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) )  e.  CC )
33 eqid 2471 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )  =  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )
3432, 33fmptd 6061 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) : ( ( 0 [,] N )  i^i  ZZ ) --> CC )
35 eqid 2471 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3635cnfldhaus 21883 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Haus
3736a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( TopOpen ` fld )  e.  Haus )
3823, 26, 28, 31, 34, 35, 37haustsms 21228 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  E* y 
y  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) ) )
3938ex 441 . . . . . . 7  |-  ( ph  ->  ( x  e.  CC  ->  E* y  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
40 moanimv 2380 . . . . . . 7  |-  ( E* y ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
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  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) ) ) )
4139, 40sylibr 217 . . . . . 6  |-  ( ph  ->  E* y ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
42 moim 2368 . . . . . 6  |-  ( A. y ( x T y  ->  ( x  e.  CC  /\  y  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
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  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )  ->  E* y  x T y ) )
4322, 41, 42sylc 61 . . . . 5  |-  ( ph  ->  E* y  x T y )
4443alrimiv 1781 . . . 4  |-  ( ph  ->  A. x E* y  x T y )
45 dffun6 5604 . . . 4  |-  ( Fun 
T  <->  ( Rel  T  /\  A. x E* y  x T y ) )
4619, 44, 45sylanbrc 677 . . 3  |-  ( ph  ->  Fun  T )
47 funfn 5618 . . 3  |-  ( Fun 
T  <->  T  Fn  dom  T )
4846, 47sylib 201 . 2  |-  ( ph  ->  T  Fn  dom  T
)
49 rnss 5069 . . . 4  |-  ( T 
C_  ( CC  X.  CC )  ->  ran  T  C_ 
ran  ( CC  X.  CC ) )
5016, 49syl 17 . . 3  |-  ( ph  ->  ran  T  C_  ran  ( CC  X.  CC ) )
51 rnxpss 5275 . . 3  |-  ran  ( CC  X.  CC )  C_  CC
5250, 51syl6ss 3430 . 2  |-  ( ph  ->  ran  T  C_  CC )
53 df-f 5593 . 2  |-  ( T : dom  T --> CC  <->  ( T  Fn  dom  T  /\  ran  T 
C_  CC ) )
5448, 52, 53sylanbrc 677 1  |-  ( ph  ->  T : dom  T --> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376   A.wal 1450    = wceq 1452    e. wcel 1904   E*wmo 2320   A.wral 2756   _Vcvv 3031    i^i cin 3389    C_ wss 3390   {csn 3959   {cpr 3961   U_ciun 4269   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ran crn 4840   Rel wrel 4844   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557    x. cmul 9562   +oocpnf 9690    - cmin 9880    / cdiv 10291   NN0cn0 10893   ZZcz 10961   [,]cicc 11663   ^cexp 12310   !cfa 12497   TopOpenctopn 15398  CMndccmn 17508   Ringcrg 17858  ℂfldccnfld 19047   TopSpctps 19996   Hauscha 20401   tsums ctsu 21218    Dncdvn 22898   Tayl ctayl 23387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-fac 12498  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cnp 20321  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tsms 21219  df-xms 21413  df-ms 21414  df-limc 22900  df-dv 22901  df-dvn 22902  df-tayl 23389
This theorem is referenced by:  tayl0  23396
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