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Theorem radcnvle 22017
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 11488 . . . . . . . 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 22014 . . . . . . . 8  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
72, 6sseldi 3461 . . . . . . 7  |-  ( ph  ->  R  e.  RR* )
87adantr 465 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR* )
9 radcnvle.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
109abscld 13039 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR )
12 0xr 9540 . . . . . . . . . . 11  |-  0  e.  RR*
13 pnfxr 11202 . . . . . . . . . . 11  |- +oo  e.  RR*
14 elicc1 11454 . . . . . . . . . . 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 1001 . . . . . . . 8  |-  ( ph  ->  0  <_  R )
18 ge0gtmnf 11254 . . . . . . . 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 9537 . . . . . . . . . 10  |-  RR  C_  RR*
2221, 10sseldi 3461 . . . . . . . . 9  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
2322adantr 465 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR* )
24 xrltle 11236 . . . . . . . 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 11251 . . . . . 6  |-  ( ( ( R  e.  RR*  /\  ( abs `  X
)  e.  RR )  /\  ( -oo  <  R  /\  R  <_  ( abs `  X ) ) )  ->  R  e.  RR )
288, 11, 20, 26, 27syl22anc 1220 . . . . 5  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR )
29 avglt1 10672 . . . . 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 3544 . . . . . . 7  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
3332, 21sstri 3472 . . . . . 6  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
3428, 11readdcld 9523 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  +  ( abs `  X
) )  e.  RR )
3534rehalfcld 10681 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  RR )
364adantr 465 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  A : NN0
--> CC )
3735recnd 9522 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  CC )
389adantr 465 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  X  e.  CC )
39 0red 9497 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  e.  RR )
4017adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  R )
4139, 28, 35, 40, 31lelttrd 9639 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
4239, 35, 41ltled 9632 . . . . . . . . . 10  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  ( ( R  +  ( abs `  X ) )  /  2 ) )
4335, 42absidd 13026 . . . . . . . . 9  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  ( ( R  +  ( abs `  X ) )  /  2 ) )  =  ( ( R  +  ( abs `  X ) )  / 
2 ) )
44 avglt2 10673 . . . . . . . . . . 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 4419 . . . . . . . 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 22012 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  seq 0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )  e.  dom  ~~>  )
51 fveq2 5798 . . . . . . . . . 10  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  ( G `  y )  =  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )
5251seqeq3d 11930 . . . . . . . . 9  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  seq 0
(  +  ,  ( G `  y ) )  =  seq 0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) ) )
5352eleq1d 2523 . . . . . . . 8  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  (  seq 0 (  +  , 
( G `  y
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( G `  (
( R  +  ( abs `  X ) )  /  2 ) ) )  e.  dom  ~~>  ) )
54 fveq2 5798 . . . . . . . . . . 11  |-  ( r  =  y  ->  ( G `  r )  =  ( G `  y ) )
5554seqeq3d 11930 . . . . . . . . . 10  |-  ( r  =  y  ->  seq 0 (  +  , 
( G `  r
) )  =  seq 0 (  +  , 
( G `  y
) ) )
5655eleq1d 2523 . . . . . . . . 9  |-  ( r  =  y  ->  (  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( G `  y
) )  e.  dom  ~~>  ) )
5756cbvrabv 3075 . . . . . . . 8  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  }  =  { y  e.  RR  |  seq 0
(  +  ,  ( G `  y ) )  e.  dom  ~~>  }
5853, 57elrab2 3224 . . . . . . 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 11397 . . . . . 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 4439 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <_  R
)
6335, 28lenltd 9630 . . . 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 9552 . . 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 965    = wceq 1370    e. wcel 1758   {crab 2802    C_ wss 3435   class class class wbr 4399    |-> cmpt 4457   dom cdm 4947   -->wf 5521   ` cfv 5525  (class class class)co 6199   supcsup 7800   CCcc 9390   RRcr 9391   0cc0 9392    + caddc 9395    x. cmul 9397   +oocpnf 9525   -oocmnf 9526   RR*cxr 9527    < clt 9528    <_ cle 9529    / cdiv 10103   2c2 10481   NN0cn0 10689   [,]cicc 11413    seqcseq 11922   ^cexp 11981   abscabs 12840    ~~> cli 13079
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-sum 13281
This theorem is referenced by:  pserdvlem2  22025  abelthlem1  22028  logtayl  22237
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