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Theorem radcnvle 20336
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 divergers 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 448 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <  ( abs `  X ) )
2 iccssxr 10993 . . . . . . . 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 20333 . . . . . . . 8  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
72, 6sseldi 3346 . . . . . . 7  |-  ( ph  ->  R  e.  RR* )
87adantr 452 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR* )
9 radcnvle.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
109abscld 12238 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
1110adantr 452 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR )
12 0xr 9131 . . . . . . . . . . 11  |-  0  e.  RR*
13 pnfxr 10713 . . . . . . . . . . 11  |-  +oo  e.  RR*
14 elicc1 10960 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR* )  ->  ( R  e.  ( 0 [,]  +oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo )
) )
1512, 13, 14mp2an 654 . . . . . . . . . 10  |-  ( R  e.  ( 0 [,] 
+oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo )
)
166, 15sylib 189 . . . . . . . . 9  |-  ( ph  ->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo ) )
1716simp2d 970 . . . . . . . 8  |-  ( ph  ->  0  <_  R )
18 ge0gtmnf 10760 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  0  <_  R )  ->  -oo  <  R )
197, 17, 18syl2anc 643 . . . . . . 7  |-  ( ph  ->  -oo  <  R )
2019adantr 452 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  -oo  <  R
)
21 ressxr 9129 . . . . . . . . . 10  |-  RR  C_  RR*
2221, 10sseldi 3346 . . . . . . . . 9  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
2322adantr 452 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR* )
24 xrltle 10742 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  ( abs `  X )  e. 
RR* )  ->  ( R  <  ( abs `  X
)  ->  R  <_  ( abs `  X ) ) )
258, 23, 24syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  ->  R  <_  ( abs `  X ) ) )
261, 25mpd 15 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <_  ( abs `  X ) )
27 xrre 10757 . . . . . 6  |-  ( ( ( R  e.  RR*  /\  ( abs `  X
)  e.  RR )  /\  (  -oo  <  R  /\  R  <_  ( abs `  X ) ) )  ->  R  e.  RR )
288, 11, 20, 26, 27syl22anc 1185 . . . . 5  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR )
29 avglt1 10205 . . . . 5  |-  ( ( R  e.  RR  /\  ( abs `  X )  e.  RR )  -> 
( R  <  ( abs `  X )  <->  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) ) )
3028, 11, 29syl2anc 643 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  <->  R  <  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )
311, 30mpbid 202 . . 3  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
32 ssrab2 3428 . . . . . . 7  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
3332, 21sstri 3357 . . . . . 6  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
3428, 11readdcld 9115 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  +  ( abs `  X
) )  e.  RR )
3534rehalfcld 10214 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  RR )
364adantr 452 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  A : NN0
--> CC )
3735recnd 9114 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  CC )
389adantr 452 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  X  e.  CC )
39 0re 9091 . . . . . . . . . . . 12  |-  0  e.  RR
4039a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  e.  RR )
4117adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  R )
4240, 28, 35, 41, 31lelttrd 9228 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
4340, 35, 42ltled 9221 . . . . . . . . . 10  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  ( ( R  +  ( abs `  X ) )  /  2 ) )
4435, 43absidd 12225 . . . . . . . . 9  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  ( ( R  +  ( abs `  X ) )  /  2 ) )  =  ( ( R  +  ( abs `  X ) )  / 
2 ) )
45 avglt2 10206 . . . . . . . . . . 11  |-  ( ( R  e.  RR  /\  ( abs `  X )  e.  RR )  -> 
( R  <  ( abs `  X )  <->  ( ( R  +  ( abs `  X ) )  / 
2 )  <  ( abs `  X ) ) )
4628, 11, 45syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  <->  ( ( R  +  ( abs `  X
) )  /  2
)  <  ( abs `  X ) ) )
471, 46mpbid 202 . . . . . . . . 9  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <  ( abs `  X ) )
4844, 47eqbrtrd 4232 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  ( ( R  +  ( abs `  X ) )  /  2 ) )  <  ( abs `  X ) )
49 radcnvle.a . . . . . . . . 9  |-  ( ph  ->  seq  0 (  +  ,  ( G `  X ) )  e. 
dom 
~~>  )
5049adantr 452 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  seq  0
(  +  ,  ( G `  X ) )  e.  dom  ~~>  )
513, 36, 37, 38, 48, 50radcnvlem3 20331 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  seq  0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )  e.  dom  ~~>  )
52 fveq2 5728 . . . . . . . . . 10  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  ( G `  y )  =  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )
5352seqeq3d 11331 . . . . . . . . 9  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  seq  0
(  +  ,  ( G `  y ) )  =  seq  0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) ) )
5453eleq1d 2502 . . . . . . . 8  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  (  seq  0 (  +  , 
( G `  y
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  (
( R  +  ( abs `  X ) )  /  2 ) ) )  e.  dom  ~~>  ) )
55 fveq2 5728 . . . . . . . . . . 11  |-  ( r  =  y  ->  ( G `  r )  =  ( G `  y ) )
5655seqeq3d 11331 . . . . . . . . . 10  |-  ( r  =  y  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  y
) ) )
5756eleq1d 2502 . . . . . . . . 9  |-  ( r  =  y  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  y
) )  e.  dom  ~~>  ) )
5857cbvrabv 2955 . . . . . . . 8  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  }  =  { y  e.  RR  |  seq  0
(  +  ,  ( G `  y ) )  e.  dom  ~~>  }
5954, 58elrab2 3094 . . . . . . 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  ~~>  ) )
6035, 51, 59sylanbrc 646 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  {
r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } )
61 supxrub 10903 . . . . . 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* ,  <  ) )
6233, 60, 61sylancr 645 . . . . 5  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <_  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  } ,  RR* ,  <  ) )
6362, 5syl6breqr 4252 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <_  R
)
6435, 28lenltd 9219 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( (
( R  +  ( abs `  X ) )  /  2 )  <_  R  <->  -.  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) ) )
6563, 64mpbid 202 . . 3  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  -.  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
6631, 65pm2.65da 560 . 2  |-  ( ph  ->  -.  R  <  ( abs `  X ) )
67 xrlenlt 9143 . . 3  |-  ( ( ( abs `  X
)  e.  RR*  /\  R  e.  RR* )  ->  (
( abs `  X
)  <_  R  <->  -.  R  <  ( abs `  X
) ) )
6822, 7, 67syl2anc 643 . 2  |-  ( ph  ->  ( ( abs `  X
)  <_  R  <->  -.  R  <  ( abs `  X
) ) )
6966, 68mpbird 224 1  |-  ( ph  ->  ( abs `  X
)  <_  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2709    C_ wss 3320   class class class wbr 4212    e. cmpt 4266   dom cdm 4878   -->wf 5450   ` cfv 5454  (class class class)co 6081   supcsup 7445   CCcc 8988   RRcr 8989   0cc0 8990    + caddc 8993    x. cmul 8995    +oocpnf 9117    -oocmnf 9118   RR*cxr 9119    < clt 9120    <_ cle 9121    / cdiv 9677   2c2 10049   NN0cn0 10221   [,]cicc 10919    seq cseq 11323   ^cexp 11382   abscabs 12039    ~~> cli 12278
This theorem is referenced by:  pserdvlem2  20344  abelthlem1  20347  logtayl  20551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480
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