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Theorem radcnv0 22007
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 0red 9491 . 2  |-  ( ph  ->  0  e.  RR )
2 nn0uz 10999 . . 3  |-  NN0  =  ( ZZ>= `  0 )
3 0zd 10762 . . 3  |-  ( ph  ->  0  e.  ZZ )
4 snfi 7493 . . . 4  |-  { 0 }  e.  Fin
54a1i 11 . . 3  |-  ( ph  ->  { 0 }  e.  Fin )
6 0nn0 10698 . . . . 5  |-  0  e.  NN0
76a1i 11 . . . 4  |-  ( ph  ->  0  e.  NN0 )
87snssd 4119 . . 3  |-  ( ph  ->  { 0 }  C_  NN0 )
9 ifid 3927 . . . 4  |-  if ( k  e.  { 0 } ,  ( ( G `  0 ) `
 k ) ,  ( ( G ` 
0 ) `  k
) )  =  ( ( G `  0
) `  k )
10 0cnd 9483 . . . . . . . 8  |-  ( ph  ->  0  e.  CC )
11 pser.g . . . . . . . . 9  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
1211pserval2 22002 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  k  e.  NN0 )  -> 
( ( G ` 
0 ) `  k
)  =  ( ( A `  k )  x.  ( 0 ^ k ) ) )
1310, 12sylan 471 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  =  ( ( A `  k
)  x.  ( 0 ^ k ) ) )
1413adantr 465 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( G `  0
) `  k )  =  ( ( A `
 k )  x.  ( 0 ^ k
) ) )
15 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
16 elnn0 10685 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
1715, 16sylib 196 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  e.  NN  \/  k  =  0 ) )
1817ord 377 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  NN  ->  k  =  0 ) )
19 elsn 3992 . . . . . . . . . . 11  |-  ( k  e.  { 0 }  <-> 
k  =  0 )
2018, 19syl6ibr 227 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  NN  ->  k  e.  { 0 } ) )
2120con1d 124 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  k  e.  { 0 }  ->  k  e.  NN ) )
2221imp 429 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  k  e.  NN )
23220expd 12134 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
0 ^ k )  =  0 )
2423oveq2d 6209 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( ( A `
 k )  x.  0 ) )
25 radcnv.a . . . . . . . . 9  |-  ( ph  ->  A : NN0 --> CC )
2625ffvelrnda 5945 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
2726adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  ( A `  k )  e.  CC )
2827mul01d 9672 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( A `  k
)  x.  0 )  =  0 )
2914, 24, 283eqtrd 2496 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  { 0 } )  ->  (
( G `  0
) `  k )  =  0 )
3029ifeq2da 3921 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  e.  { 0 } ,  ( ( G `  0 ) `
 k ) ,  ( ( G ` 
0 ) `  k
) )  =  if ( k  e.  {
0 } ,  ( ( G `  0
) `  k ) ,  0 ) )
319, 30syl5eqr 2506 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  =  if ( k  e.  {
0 } ,  ( ( G `  0
) `  k ) ,  0 ) )
328sselda 3457 . . . 4  |-  ( (
ph  /\  k  e.  { 0 } )  -> 
k  e.  NN0 )
3311, 25, 10psergf 22003 . . . . 5  |-  ( ph  ->  ( G `  0
) : NN0 --> CC )
3433ffvelrnda 5945 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  0 ) `  k )  e.  CC )
3532, 34syldan 470 . . 3  |-  ( (
ph  /\  k  e.  { 0 } )  -> 
( ( G ` 
0 ) `  k
)  e.  CC )
362, 3, 5, 8, 31, 35fsumcvg3 13317 . 2  |-  ( ph  ->  seq 0 (  +  ,  ( G ` 
0 ) )  e. 
dom 
~~>  )
37 fveq2 5792 . . . . 5  |-  ( r  =  0  ->  ( G `  r )  =  ( G ` 
0 ) )
3837seqeq3d 11924 . . . 4  |-  ( r  =  0  ->  seq 0 (  +  , 
( G `  r
) )  =  seq 0 (  +  , 
( G `  0
) ) )
3938eleq1d 2520 . . 3  |-  ( r  =  0  ->  (  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( G `  0
) )  e.  dom  ~~>  ) )
4039elrab 3217 . 2  |-  ( 0  e.  { r  e.  RR  |  seq 0
(  +  ,  ( G `  r ) )  e.  dom  ~~>  }  <->  ( 0  e.  RR  /\  seq 0 (  +  , 
( G `  0
) )  e.  dom  ~~>  ) )
411, 36, 40sylanbrc 664 1  |-  ( ph  ->  0  e.  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2799   ifcif 3892   {csn 3978    |-> cmpt 4451   dom cdm 4941   -->wf 5515   ` cfv 5519  (class class class)co 6193   Fincfn 7413   CCcc 9384   RRcr 9385   0cc0 9386    + caddc 9389    x. cmul 9391   NNcn 10426   NN0cn0 10683    seqcseq 11916   ^cexp 11975    ~~> cli 13073
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 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
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 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077
This theorem is referenced by:  radcnvcl  22008
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