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Theorem radcnv0 20199
Description: Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
Hypotheses
Ref Expression
pser.g  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
radcnv.a  |-  ( ph  ->  A : NN0 --> CC )
Assertion
Ref Expression
radcnv0  |-  ( ph  ->  0  e.  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } )
Distinct variable groups:    x, n, A    G, r
Allowed substitution hints:    ph( x, n, r)    A( r)    G( x, n)

Proof of Theorem radcnv0
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 0re 9024 . . 3  |-  0  e.  RR
21a1i 11 . 2  |-  ( ph  ->  0  e.  RR )
3 nn0uz 10452 . . 3  |-  NN0  =  ( ZZ>= `  0 )
4 0z 10225 . . . 4  |-  0  e.  ZZ
54a1i 11 . . 3  |-  ( ph  ->  0  e.  ZZ )
6 snfi 7123 . . . 4  |-  { 0 }  e.  Fin
76a1i 11 . . 3  |-  ( ph  ->  { 0 }  e.  Fin )
8 0nn0 10168 . . . . 5  |-  0  e.  NN0
98a1i 11 . . . 4  |-  ( ph  ->  0  e.  NN0 )
109snssd 3886 . . 3  |-  ( ph  ->  { 0 }  C_  NN0 )
11 ifid 3714 . . . 4  |-  if ( k  e.  { 0 } ,  ( ( G `  0 ) `
 k ) ,  ( ( G ` 
0 ) `  k
) )  =  ( ( G `  0
) `  k )
12 0cn 9017 . . . . . . . . 9  |-  0  e.  CC
1312a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  CC )
14 pser.g . . . . . . . . 9  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
1514pserval2 20194 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  k  e.  NN0 )  -> 
( ( G ` 
0 ) `  k
)  =  ( ( A `  k )  x.  ( 0 ^ k ) ) )
1613, 15sylan 458 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  =  ( ( A `  k
)  x.  ( 0 ^ k ) ) )
1716adantr 452 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( G `  0
) `  k )  =  ( ( A `
 k )  x.  ( 0 ^ k
) ) )
18 simpr 448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
19 elnn0 10155 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
2018, 19sylib 189 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  e.  NN  \/  k  =  0 ) )
2120ord 367 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  NN  ->  k  =  0 ) )
22 elsn 3772 . . . . . . . . . . 11  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2321, 22syl6ibr 219 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  NN  ->  k  e.  { 0 } ) )
2423con1d 118 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  { 0 }  ->  k  e.  NN ) )
2524imp 419 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  k  e.  NN )
26250expd 11466 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
0 ^ k )  =  0 )
2726oveq2d 6036 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( ( A `
 k )  x.  0 ) )
28 radcnv.a . . . . . . . . 9  |-  ( ph  ->  A : NN0 --> CC )
2928ffvelrnda 5809 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
3029adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  ( A `  k )  e.  CC )
3130mul01d 9197 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( A `  k
)  x.  0 )  =  0 )
3217, 27, 313eqtrd 2423 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( G `  0
) `  k )  =  0 )
3332ifeq2da 3708 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  e.  { 0 } ,  ( ( G `  0 ) `
 k ) ,  ( ( G ` 
0 ) `  k
) )  =  if ( k  e.  {
0 } ,  ( ( G `  0
) `  k ) ,  0 ) )
3411, 33syl5eqr 2433 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  =  if ( k  e.  {
0 } ,  ( ( G `  0
) `  k ) ,  0 ) )
3510sselda 3291 . . . 4  |-  ( (
ph  /\  k  e.  { 0 } )  -> 
k  e.  NN0 )
3614, 28, 13psergf 20195 . . . . 5  |-  ( ph  ->  ( G `  0
) : NN0 --> CC )
3736ffvelrnda 5809 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  e.  CC )
3835, 37syldan 457 . . 3  |-  ( (
ph  /\  k  e.  { 0 } )  -> 
( ( G ` 
0 ) `  k
)  e.  CC )
393, 5, 7, 10, 34, 38fsumcvg3 12450 . 2  |-  ( ph  ->  seq  0 (  +  ,  ( G ` 
0 ) )  e. 
dom 
~~>  )
40 fveq2 5668 . . . . 5  |-  ( r  =  0  ->  ( G `  r )  =  ( G ` 
0 ) )
4140seqeq3d 11258 . . . 4  |-  ( r  =  0  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  0
) ) )
4241eleq1d 2453 . . 3  |-  ( r  =  0  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  0
) )  e.  dom  ~~>  ) )
4342elrab 3035 . 2  |-  ( 0  e.  { r  e.  RR  |  seq  0
(  +  ,  ( G `  r ) )  e.  dom  ~~>  }  <->  ( 0  e.  RR  /\  seq  0 (  +  , 
( G `  0
) )  e.  dom  ~~>  ) )
442, 39, 43sylanbrc 646 1  |-  ( ph  ->  0  e.  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2653   ifcif 3682   {csn 3757    e. cmpt 4207   dom cdm 4818   -->wf 5390   ` cfv 5394  (class class class)co 6020   Fincfn 7045   CCcc 8921   RRcr 8922   0cc0 8923    + caddc 8926    x. cmul 8928   NNcn 9932   NN0cn0 10153   ZZcz 10214    seq cseq 11250   ^cexp 11309    ~~> cli 12205
This theorem is referenced by:  radcnvcl  20200
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209
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