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Theorem radcnvle 22546
Description: If  X is a convergent point of the infinite series, then  X is within the closed disk of radius  R centered at zero. Or, by contraposition, the series diverges at any point strictly more than  R from the origin. (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 )
radcnv.r  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
radcnvle.x  |-  ( ph  ->  X  e.  CC )
radcnvle.a  |-  ( ph  ->  seq 0 (  +  ,  ( G `  X ) )  e. 
dom 
~~>  )
Assertion
Ref Expression
radcnvle  |-  ( ph  ->  ( abs `  X
)  <_  R )
Distinct variable groups:    x, n, A    G, r
Allowed substitution hints:    ph( x, n, r)    A( r)    R( x, n, r)    G( x, n)    X( x, n, r)

Proof of Theorem radcnvle
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <  ( abs `  X ) )
2 iccssxr 11603 . . . . . . . 8  |-  ( 0 [,] +oo )  C_  RR*
3 pser.g . . . . . . . . 9  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
4 radcnv.a . . . . . . . . 9  |-  ( ph  ->  A : NN0 --> CC )
5 radcnv.r . . . . . . . . 9  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
63, 4, 5radcnvcl 22543 . . . . . . . 8  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
72, 6sseldi 3502 . . . . . . 7  |-  ( ph  ->  R  e.  RR* )
87adantr 465 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR* )
9 radcnvle.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
109abscld 13223 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR )
12 0xr 9636 . . . . . . . . . . 11  |-  0  e.  RR*
13 pnfxr 11317 . . . . . . . . . . 11  |- +oo  e.  RR*
14 elicc1 11569 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  ( R  e.  ( 0 [,] +oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_ +oo )
) )
1512, 13, 14mp2an 672 . . . . . . . . . 10  |-  ( R  e.  ( 0 [,] +oo )  <->  ( R  e. 
RR*  /\  0  <_  R  /\  R  <_ +oo )
)
166, 15sylib 196 . . . . . . . . 9  |-  ( ph  ->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_ +oo ) )
1716simp2d 1009 . . . . . . . 8  |-  ( ph  ->  0  <_  R )
18 ge0gtmnf 11369 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  0  <_  R )  -> -oo  <  R )
197, 17, 18syl2anc 661 . . . . . . 7  |-  ( ph  -> -oo  <  R )
2019adantr 465 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  -> -oo  <  R
)
21 ressxr 9633 . . . . . . . . . 10  |-  RR  C_  RR*
2221, 10sseldi 3502 . . . . . . . . 9  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
2322adantr 465 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR* )
24 xrltle 11351 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  ( abs `  X )  e. 
RR* )  ->  ( R  <  ( abs `  X
)  ->  R  <_  ( abs `  X ) ) )
258, 23, 24syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  ->  R  <_  ( abs `  X ) ) )
261, 25mpd 15 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <_  ( abs `  X ) )
27 xrre 11366 . . . . . 6  |-  ( ( ( R  e.  RR*  /\  ( abs `  X
)  e.  RR )  /\  ( -oo  <  R  /\  R  <_  ( abs `  X ) ) )  ->  R  e.  RR )
288, 11, 20, 26, 27syl22anc 1229 . . . . 5  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR )
29 avglt1 10772 . . . . 5  |-  ( ( R  e.  RR  /\  ( abs `  X )  e.  RR )  -> 
( R  <  ( abs `  X )  <->  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) ) )
3028, 11, 29syl2anc 661 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  <->  R  <  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )
311, 30mpbid 210 . . 3  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
32 ssrab2 3585 . . . . . . 7  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
3332, 21sstri 3513 . . . . . 6  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
3428, 11readdcld 9619 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  +  ( abs `  X
) )  e.  RR )
3534rehalfcld 10781 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  RR )
364adantr 465 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  A : NN0
--> CC )
3735recnd 9618 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  CC )
389adantr 465 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  X  e.  CC )
39 0red 9593 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  e.  RR )
4017adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  R )
4139, 28, 35, 40, 31lelttrd 9735 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
4239, 35, 41ltled 9728 . . . . . . . . . 10  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  ( ( R  +  ( abs `  X ) )  /  2 ) )
4335, 42absidd 13210 . . . . . . . . 9  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  ( ( R  +  ( abs `  X ) )  /  2 ) )  =  ( ( R  +  ( abs `  X ) )  / 
2 ) )
44 avglt2 10773 . . . . . . . . . . 11  |-  ( ( R  e.  RR  /\  ( abs `  X )  e.  RR )  -> 
( R  <  ( abs `  X )  <->  ( ( R  +  ( abs `  X ) )  / 
2 )  <  ( abs `  X ) ) )
4528, 11, 44syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  <->  ( ( R  +  ( abs `  X
) )  /  2
)  <  ( abs `  X ) ) )
461, 45mpbid 210 . . . . . . . . 9  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <  ( abs `  X ) )
4743, 46eqbrtrd 4467 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  ( ( R  +  ( abs `  X ) )  /  2 ) )  <  ( abs `  X ) )
48 radcnvle.a . . . . . . . . 9  |-  ( ph  ->  seq 0 (  +  ,  ( G `  X ) )  e. 
dom 
~~>  )
4948adantr 465 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  seq 0
(  +  ,  ( G `  X ) )  e.  dom  ~~>  )
503, 36, 37, 38, 47, 49radcnvlem3 22541 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  seq 0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )  e.  dom  ~~>  )
51 fveq2 5864 . . . . . . . . . 10  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  ( G `  y )  =  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )
5251seqeq3d 12078 . . . . . . . . 9  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  seq 0
(  +  ,  ( G `  y ) )  =  seq 0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) ) )
5352eleq1d 2536 . . . . . . . 8  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  (  seq 0 (  +  , 
( G `  y
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( G `  (
( R  +  ( abs `  X ) )  /  2 ) ) )  e.  dom  ~~>  ) )
54 fveq2 5864 . . . . . . . . . . 11  |-  ( r  =  y  ->  ( G `  r )  =  ( G `  y ) )
5554seqeq3d 12078 . . . . . . . . . 10  |-  ( r  =  y  ->  seq 0 (  +  , 
( G `  r
) )  =  seq 0 (  +  , 
( G `  y
) ) )
5655eleq1d 2536 . . . . . . . . 9  |-  ( r  =  y  ->  (  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( G `  y
) )  e.  dom  ~~>  ) )
5756cbvrabv 3112 . . . . . . . 8  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  }  =  { y  e.  RR  |  seq 0
(  +  ,  ( G `  y ) )  e.  dom  ~~>  }
5853, 57elrab2 3263 . . . . . . 7  |-  ( ( ( R  +  ( abs `  X ) )  /  2 )  e.  { r  e.  RR  |  seq 0
(  +  ,  ( G `  r ) )  e.  dom  ~~>  }  <->  ( (
( R  +  ( abs `  X ) )  /  2 )  e.  RR  /\  seq 0 (  +  , 
( G `  (
( R  +  ( abs `  X ) )  /  2 ) ) )  e.  dom  ~~>  ) )
5935, 50, 58sylanbrc 664 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  {
r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } )
60 supxrub 11512 . . . . . 6  |-  ( ( { r  e.  RR  |  seq 0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  }  C_  RR*  /\  (
( R  +  ( abs `  X ) )  /  2 )  e.  { r  e.  RR  |  seq 0
(  +  ,  ( G `  r ) )  e.  dom  ~~>  } )  ->  ( ( R  +  ( abs `  X
) )  /  2
)  <_  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  } ,  RR* ,  <  ) )
6133, 59, 60sylancr 663 . . . . 5  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <_  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  } ,  RR* ,  <  ) )
6261, 5syl6breqr 4487 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <_  R
)
6335, 28lenltd 9726 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( (
( R  +  ( abs `  X ) )  /  2 )  <_  R  <->  -.  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) ) )
6462, 63mpbid 210 . . 3  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  -.  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
6531, 64pm2.65da 576 . 2  |-  ( ph  ->  -.  R  <  ( abs `  X ) )
66 xrlenlt 9648 . . 3  |-  ( ( ( abs `  X
)  e.  RR*  /\  R  e.  RR* )  ->  (
( abs `  X
)  <_  R  <->  -.  R  <  ( abs `  X
) ) )
6722, 7, 66syl2anc 661 . 2  |-  ( ph  ->  ( ( abs `  X
)  <_  R  <->  -.  R  <  ( abs `  X
) ) )
6865, 67mpbird 232 1  |-  ( ph  ->  ( abs `  X
)  <_  R )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   -->wf 5582   ` cfv 5586  (class class class)co 6282   supcsup 7896   CCcc 9486   RRcr 9487   0cc0 9488    + caddc 9491    x. cmul 9493   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623    < clt 9624    <_ cle 9625    / cdiv 10202   2c2 10581   NN0cn0 10791   [,]cicc 11528    seqcseq 12070   ^cexp 12129   abscabs 13024    ~~> cli 13263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-limsup 13250  df-clim 13267  df-rlim 13268  df-sum 13465
This theorem is referenced by:  pserdvlem2  22554  abelthlem1  22557  logtayl  22766
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