MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  taylf Structured version   Unicode version

Theorem taylf 22628
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 22626 . . . . . 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 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
98snssd 4160 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
101, 2, 3, 4, 5taylfvallem 22625 . . . . . . . . 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 5098 . . . . . . . . 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 661 . . . . . . . 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 2857 . . . . . . 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 4356 . . . . . . 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 212 . . . . . 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 3523 . . . . 5  |-  ( ph  ->  T  C_  ( CC  X.  CC ) )
17 relxp 5100 . . . . 5  |-  Rel  ( CC  X.  CC )
18 relss 5080 . . . . 5  |-  ( T 
C_  ( CC  X.  CC )  ->  ( Rel  ( CC  X.  CC )  ->  Rel  T )
)
1916, 17, 18mpisyl 18 . . . 4  |-  ( ph  ->  Rel  T )
201, 2, 3, 4, 5, 6eltayl 22627 . . . . . . . 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 207 . . . . . . 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 1706 . . . . . 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 18298 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
24 cnring 18314 . . . . . . . . . 10  |-fld  e.  Ring
25 ringcmn 17103 . . . . . . . . . 10  |-  (fld  e.  Ring  ->fld  e. CMnd )
2624, 25mp1i 12 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
27 cnfldtps 21158 . . . . . . . . . 10  |-fld  e.  TopSp
2827a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
29 ovex 6309 . . . . . . . . . . 11  |-  ( 0 [,] N )  e. 
_V
3029inex1 4578 . . . . . . . . . 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 22624 . . . . . . . . . 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 2443 . . . . . . . . . 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 6040 . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3635cnfldhaus 21165 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Haus
3736a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( TopOpen ` fld )  e.  Haus )
3823, 26, 28, 31, 34, 35, 37haustsms 20507 . . . . . . . 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 434 . . . . . . 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 2338 . . . . . . 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 212 . . . . . 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 2325 . . . . . 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 60 . . . . 5  |-  ( ph  ->  E* y  x T y )
4443alrimiv 1706 . . . 4  |-  ( ph  ->  A. x E* y  x T y )
45 dffun6 5593 . . . 4  |-  ( Fun 
T  <->  ( Rel  T  /\  A. x E* y  x T y ) )
4619, 44, 45sylanbrc 664 . . 3  |-  ( ph  ->  Fun  T )
47 funfn 5607 . . 3  |-  ( Fun 
T  <->  T  Fn  dom  T )
4846, 47sylib 196 . 2  |-  ( ph  ->  T  Fn  dom  T
)
49 rnss 5221 . . . 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 5429 . . 3  |-  ran  ( CC  X.  CC )  C_  CC
5250, 51syl6ss 3501 . 2  |-  ( ph  ->  ran  T  C_  CC )
53 df-f 5582 . 2  |-  ( T : dom  T --> CC  <->  ( T  Fn  dom  T  /\  ran  T 
C_  CC ) )
5448, 52, 53sylanbrc 664 1  |-  ( ph  ->  T : dom  T --> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1381    = wceq 1383    e. wcel 1804   E*wmo 2269   A.wral 2793   _Vcvv 3095    i^i cin 3460    C_ wss 3461   {csn 4014   {cpr 4016   U_ciun 4315   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   dom cdm 4989   ran crn 4990   Rel wrel 4994   Fun wfun 5572    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495    x. cmul 9500   +oocpnf 9628    - cmin 9810    / cdiv 10212   NN0cn0 10801   ZZcz 10870   [,]cicc 11541   ^cexp 12145   !cfa 12332   TopOpenctopn 14696  CMndccmn 16672   Ringcrg 17072  ℂfldccnfld 18294   TopSpctps 19270   Hauscha 19682   tsums ctsu 20497    Dncdvn 22141   Tayl ctayl 22620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-icc 11545  df-fz 11682  df-fzo 11804  df-seq 12087  df-exp 12146  df-fac 12333  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-mulr 14588  df-starv 14589  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-cntz 16229  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cnp 19602  df-haus 19689  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-tsms 20498  df-xms 20696  df-ms 20697  df-limc 22143  df-dv 22144  df-dvn 22145  df-tayl 22622
This theorem is referenced by:  tayl0  22629
  Copyright terms: Public domain W3C validator