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Theorem radcnv0 19792
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 8838 . . 3  |-  0  e.  RR
21a1i 10 . 2  |-  ( ph  ->  0  e.  RR )
3 nn0uz 10262 . . 3  |-  NN0  =  ( ZZ>= `  0 )
4 0z 10035 . . . 4  |-  0  e.  ZZ
54a1i 10 . . 3  |-  ( ph  ->  0  e.  ZZ )
6 snfi 6941 . . . 4  |-  { 0 }  e.  Fin
76a1i 10 . . 3  |-  ( ph  ->  { 0 }  e.  Fin )
8 0nn0 9980 . . . . 5  |-  0  e.  NN0
98a1i 10 . . . 4  |-  ( ph  ->  0  e.  NN0 )
109snssd 3760 . . 3  |-  ( ph  ->  { 0 }  C_  NN0 )
11 ifid 3597 . . . 4  |-  if ( k  e.  { 0 } ,  ( ( G `  0 ) `
 k ) ,  ( ( G ` 
0 ) `  k
) )  =  ( ( G `  0
) `  k )
12 0cn 8831 . . . . . . . . 9  |-  0  e.  CC
1312a1i 10 . . . . . . . 8  |-  ( ph  ->  0  e.  CC )
14 pser.g . . . . . . . . 9  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
1514pserval2 19787 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  k  e.  NN0 )  -> 
( ( G ` 
0 ) `  k
)  =  ( ( A `  k )  x.  ( 0 ^ k ) ) )
1613, 15sylan 457 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  =  ( ( A `  k
)  x.  ( 0 ^ k ) ) )
1716adantr 451 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( G `  0
) `  k )  =  ( ( A `
 k )  x.  ( 0 ^ k
) ) )
18 simpr 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
19 elnn0 9967 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
2018, 19sylib 188 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  e.  NN  \/  k  =  0 ) )
2120ord 366 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  NN  ->  k  =  0 ) )
22 elsn 3655 . . . . . . . . . . 11  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2321, 22syl6ibr 218 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  NN  ->  k  e.  { 0 } ) )
2423con1d 116 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  { 0 }  ->  k  e.  NN ) )
2524imp 418 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  k  e.  NN )
26250expd 11261 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
0 ^ k )  =  0 )
2726oveq2d 5874 . . . . . 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 )
29 ffvelrn 5663 . . . . . . . . 9  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
3028, 29sylan 457 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
3130adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  ( A `  k )  e.  CC )
3231mul01d 9011 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( A `  k
)  x.  0 )  =  0 )
3317, 27, 323eqtrd 2319 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( G `  0
) `  k )  =  0 )
3433ifeq2da 3591 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  e.  { 0 } ,  ( ( G `  0 ) `
 k ) ,  ( ( G ` 
0 ) `  k
) )  =  if ( k  e.  {
0 } ,  ( ( G `  0
) `  k ) ,  0 ) )
3511, 34syl5eqr 2329 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  =  if ( k  e.  {
0 } ,  ( ( G `  0
) `  k ) ,  0 ) )
3610sselda 3180 . . . 4  |-  ( (
ph  /\  k  e.  { 0 } )  -> 
k  e.  NN0 )
3714, 28, 13psergf 19788 . . . . 5  |-  ( ph  ->  ( G `  0
) : NN0 --> CC )
38 ffvelrn 5663 . . . . 5  |-  ( ( ( G `  0
) : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( G ` 
0 ) `  k
)  e.  CC )
3937, 38sylan 457 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  e.  CC )
4036, 39syldan 456 . . 3  |-  ( (
ph  /\  k  e.  { 0 } )  -> 
( ( G ` 
0 ) `  k
)  e.  CC )
413, 5, 7, 10, 35, 40fsumcvg3 12202 . 2  |-  ( ph  ->  seq  0 (  +  ,  ( G ` 
0 ) )  e. 
dom 
~~>  )
42 fveq2 5525 . . . . 5  |-  ( r  =  0  ->  ( G `  r )  =  ( G ` 
0 ) )
4342seqeq3d 11054 . . . 4  |-  ( r  =  0  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  0
) ) )
4443eleq1d 2349 . . 3  |-  ( r  =  0  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  0
) )  e.  dom  ~~>  ) )
4544elrab 2923 . 2  |-  ( 0  e.  { r  e.  RR  |  seq  0
(  +  ,  ( G `  r ) )  e.  dom  ~~>  }  <->  ( 0  e.  RR  /\  seq  0 (  +  , 
( G `  0
) )  e.  dom  ~~>  ) )
462, 41, 45sylanbrc 645 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 357    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   ifcif 3565   {csn 3640    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742   NNcn 9746   NN0cn0 9965   ZZcz 10024    seq cseq 11046   ^cexp 11104    ~~> cli 11958
This theorem is referenced by:  radcnvcl  19793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
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