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

Theorem taylf 21942
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 21940 . . . . . 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 4116 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
101, 2, 3, 4, 5taylfvallem 21939 . . . . . . . . 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 5043 . . . . . . . . 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 2822 . . . . . . 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 4309 . . . . . . 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 3488 . . . . 5  |-  ( ph  ->  T  C_  ( CC  X.  CC ) )
17 relxp 5045 . . . . 5  |-  Rel  ( CC  X.  CC )
18 relss 5025 . . . . 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 21941 . . . . . . . 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 1686 . . . . . 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 17931 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
24 cnrng 17947 . . . . . . . . . 10  |-fld  e.  Ring
25 rngcmn 16781 . . . . . . . . . 10  |-  (fld  e.  Ring  ->fld  e. CMnd )
2624, 25mp1i 12 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
27 cnfldtps 20473 . . . . . . . . . 10  |-fld  e.  TopSp
2827a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
29 ovex 6215 . . . . . . . . . . 11  |-  ( 0 [,] N )  e. 
_V
3029inex1 4531 . . . . . . . . . 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 21938 . . . . . . . . . 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 2451 . . . . . . . . . 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 5966 . . . . . . . . 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 2451 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3635cnfldhaus 20480 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Haus
3736a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( TopOpen ` fld )  e.  Haus )
3823, 26, 28, 31, 34, 35, 37haustsms 19822 . . . . . . . 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 2341 . . . . . . 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 2326 . . . . . 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 1686 . . . 4  |-  ( ph  ->  A. x E* y  x T y )
45 dffun6 5531 . . . 4  |-  ( Fun 
T  <->  ( Rel  T  /\  A. x E* y  x T y ) )
4619, 44, 45sylanbrc 664 . . 3  |-  ( ph  ->  Fun  T )
47 funfn 5545 . . 3  |-  ( Fun 
T  <->  T  Fn  dom  T )
4846, 47sylib 196 . 2  |-  ( ph  ->  T  Fn  dom  T
)
49 rnss 5166 . . . 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 5368 . . 3  |-  ran  ( CC  X.  CC )  C_  CC
5250, 51syl6ss 3466 . 2  |-  ( ph  ->  ran  T  C_  CC )
53 df-f 5520 . 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 1368    = wceq 1370    e. wcel 1758   E*wmo 2261   A.wral 2795   _Vcvv 3068    i^i cin 3425    C_ wss 3426   {csn 3975   {cpr 3977   U_ciun 4269   class class class wbr 4390    |-> cmpt 4448    X. cxp 4936   dom cdm 4938   ran crn 4939   Rel wrel 4943   Fun wfun 5510    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383    x. cmul 9388   +oocpnf 9516    - cmin 9696    / cdiv 10094   NN0cn0 10680   ZZcz 10747   [,]cicc 11404   ^cexp 11966   !cfa 12152   TopOpenctopn 14462  CMndccmn 16381   Ringcrg 16751  ℂfldccnfld 17927   TopSpctps 18617   Hauscha 19028   tsums ctsu 19812    Dncdvn 21455   Tayl ctayl 21934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-icc 11408  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-fac 12153  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-plusg 14353  df-mulr 14354  df-starv 14355  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-mnd 15517  df-grp 15647  df-minusg 15648  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-cring 16754  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-fbas 17923  df-fg 17924  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cld 18739  df-ntr 18740  df-cls 18741  df-nei 18818  df-lp 18856  df-perf 18857  df-cnp 18948  df-haus 19035  df-fil 19535  df-fm 19627  df-flim 19628  df-flf 19629  df-tsms 19813  df-xms 20011  df-ms 20012  df-limc 21457  df-dv 21458  df-dvn 21459  df-tayl 21936
This theorem is referenced by:  tayl0  21943
  Copyright terms: Public domain W3C validator