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Theorem radcnv0 19808
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 8854 . . 3  |-  0  e.  RR
21a1i 10 . 2  |-  ( ph  ->  0  e.  RR )
3 nn0uz 10278 . . 3  |-  NN0  =  ( ZZ>= `  0 )
4 0z 10051 . . . 4  |-  0  e.  ZZ
54a1i 10 . . 3  |-  ( ph  ->  0  e.  ZZ )
6 snfi 6957 . . . 4  |-  { 0 }  e.  Fin
76a1i 10 . . 3  |-  ( ph  ->  { 0 }  e.  Fin )
8 0nn0 9996 . . . . 5  |-  0  e.  NN0
98a1i 10 . . . 4  |-  ( ph  ->  0  e.  NN0 )
109snssd 3776 . . 3  |-  ( ph  ->  { 0 }  C_  NN0 )
11 ifid 3610 . . . 4  |-  if ( k  e.  { 0 } ,  ( ( G `  0 ) `
 k ) ,  ( ( G ` 
0 ) `  k
) )  =  ( ( G `  0
) `  k )
12 0cn 8847 . . . . . . . . 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 19803 . . . . . . . 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 9983 . . . . . . . . . . . . 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 3668 . . . . . . . . . . 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 11277 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
0 ^ k )  =  0 )
2726oveq2d 5890 . . . . . 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 5679 . . . . . . . . 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 9027 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( A `  k
)  x.  0 )  =  0 )
3317, 27, 323eqtrd 2332 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( G `  0
) `  k )  =  0 )
3433ifeq2da 3604 . . . 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 2342 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  =  if ( k  e.  {
0 } ,  ( ( G `  0
) `  k ) ,  0 ) )
3610sselda 3193 . . . 4  |-  ( (
ph  /\  k  e.  { 0 } )  -> 
k  e.  NN0 )
3714, 28, 13psergf 19804 . . . . 5  |-  ( ph  ->  ( G `  0
) : NN0 --> CC )
38 ffvelrn 5679 . . . . 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 12218 . 2  |-  ( ph  ->  seq  0 (  +  ,  ( G ` 
0 ) )  e. 
dom 
~~>  )
42 fveq2 5541 . . . . 5  |-  ( r  =  0  ->  ( G `  r )  =  ( G ` 
0 ) )
4342seqeq3d 11070 . . . 4  |-  ( r  =  0  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  0
) ) )
4443eleq1d 2362 . . 3  |-  ( r  =  0  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  0
) )  e.  dom  ~~>  ) )
4544elrab 2936 . 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 1632    e. wcel 1696   {crab 2560   ifcif 3578   {csn 3653    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758   NNcn 9762   NN0cn0 9981   ZZcz 10040    seq cseq 11062   ^cexp 11120    ~~> cli 11974
This theorem is referenced by:  radcnvcl  19809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978
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