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Theorem radcnvle 19812
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 447 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <  ( abs `  X ) )
2 iccssxr 10748 . . . . . . . 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 19809 . . . . . . . 8  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
72, 6sseldi 3191 . . . . . . 7  |-  ( ph  ->  R  e.  RR* )
87adantr 451 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR* )
9 radcnvle.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
109abscld 11934 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
1110adantr 451 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR )
12 0xr 8894 . . . . . . . . . . 11  |-  0  e.  RR*
13 pnfxr 10471 . . . . . . . . . . 11  |-  +oo  e.  RR*
14 elicc1 10716 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR* )  ->  ( R  e.  ( 0 [,]  +oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo )
) )
1512, 13, 14mp2an 653 . . . . . . . . . 10  |-  ( R  e.  ( 0 [,] 
+oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo )
)
166, 15sylib 188 . . . . . . . . 9  |-  ( ph  ->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo ) )
1716simp2d 968 . . . . . . . 8  |-  ( ph  ->  0  <_  R )
18 ge0gtmnf 10517 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  0  <_  R )  ->  -oo  <  R )
197, 17, 18syl2anc 642 . . . . . . 7  |-  ( ph  ->  -oo  <  R )
2019adantr 451 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  -oo  <  R
)
21 ressxr 8892 . . . . . . . . . 10  |-  RR  C_  RR*
2221, 10sseldi 3191 . . . . . . . . 9  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
2322adantr 451 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR* )
24 xrltle 10499 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  ( abs `  X )  e. 
RR* )  ->  ( R  <  ( abs `  X
)  ->  R  <_  ( abs `  X ) ) )
258, 23, 24syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  ->  R  <_  ( abs `  X ) ) )
261, 25mpd 14 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <_  ( abs `  X ) )
27 xrre 10514 . . . . . 6  |-  ( ( ( R  e.  RR*  /\  ( abs `  X
)  e.  RR )  /\  (  -oo  <  R  /\  R  <_  ( abs `  X ) ) )  ->  R  e.  RR )
288, 11, 20, 26, 27syl22anc 1183 . . . . 5  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR )
29 avglt1 9965 . . . . 5  |-  ( ( R  e.  RR  /\  ( abs `  X )  e.  RR )  -> 
( R  <  ( abs `  X )  <->  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) ) )
3028, 11, 29syl2anc 642 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  <->  R  <  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )
311, 30mpbid 201 . . 3  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
32 ssrab2 3271 . . . . . . 7  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
3332, 21sstri 3201 . . . . . 6  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
3428, 11readdcld 8878 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  +  ( abs `  X
) )  e.  RR )
3534rehalfcld 9974 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  RR )
364adantr 451 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  A : NN0
--> CC )
3735recnd 8877 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  CC )
389adantr 451 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  X  e.  CC )
39 0re 8854 . . . . . . . . . . . 12  |-  0  e.  RR
4039a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  e.  RR )
4117adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  R )
4240, 28, 35, 41, 31lelttrd 8990 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
4340, 35, 42ltled 8983 . . . . . . . . . 10  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  ( ( R  +  ( abs `  X ) )  /  2 ) )
4435, 43absidd 11921 . . . . . . . . 9  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  ( ( R  +  ( abs `  X ) )  /  2 ) )  =  ( ( R  +  ( abs `  X ) )  / 
2 ) )
45 avglt2 9966 . . . . . . . . . . 11  |-  ( ( R  e.  RR  /\  ( abs `  X )  e.  RR )  -> 
( R  <  ( abs `  X )  <->  ( ( R  +  ( abs `  X ) )  / 
2 )  <  ( abs `  X ) ) )
4628, 11, 45syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  <  ( abs `  X
)  <->  ( ( R  +  ( abs `  X
) )  /  2
)  <  ( abs `  X ) ) )
471, 46mpbid 201 . . . . . . . . 9  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <  ( abs `  X ) )
4844, 47eqbrtrd 4059 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  ( ( R  +  ( abs `  X ) )  /  2 ) )  <  ( abs `  X ) )
49 radcnvle.a . . . . . . . . 9  |-  ( ph  ->  seq  0 (  +  ,  ( G `  X ) )  e. 
dom 
~~>  )
5049adantr 451 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  seq  0
(  +  ,  ( G `  X ) )  e.  dom  ~~>  )
513, 36, 37, 38, 48, 50radcnvlem3 19807 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  seq  0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )  e.  dom  ~~>  )
52 fveq2 5541 . . . . . . . . . 10  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  ( G `  y )  =  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )
5352seqeq3d 11070 . . . . . . . . 9  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  seq  0
(  +  ,  ( G `  y ) )  =  seq  0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) ) )
5453eleq1d 2362 . . . . . . . 8  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  (  seq  0 (  +  , 
( G `  y
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  (
( R  +  ( abs `  X ) )  /  2 ) ) )  e.  dom  ~~>  ) )
55 fveq2 5541 . . . . . . . . . . 11  |-  ( r  =  y  ->  ( G `  r )  =  ( G `  y ) )
5655seqeq3d 11070 . . . . . . . . . 10  |-  ( r  =  y  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  y
) ) )
5756eleq1d 2362 . . . . . . . . 9  |-  ( r  =  y  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  y
) )  e.  dom  ~~>  ) )
5857cbvrabv 2800 . . . . . . . 8  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  }  =  { y  e.  RR  |  seq  0
(  +  ,  ( G `  y ) )  e.  dom  ~~>  }
5954, 58elrab2 2938 . . . . . . 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 645 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  {
r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } )
61 supxrub 10659 . . . . . 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 644 . . . . 5  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <_  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  } ,  RR* ,  <  ) )
6362, 5syl6breqr 4079 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <_  R
)
6435, 28lenltd 8981 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( (
( R  +  ( abs `  X ) )  /  2 )  <_  R  <->  -.  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) ) )
6563, 64mpbid 201 . . 3  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  -.  R  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
6631, 65pm2.65da 559 . 2  |-  ( ph  ->  -.  R  <  ( abs `  X ) )
67 xrlenlt 8906 . . 3  |-  ( ( ( abs `  X
)  e.  RR*  /\  R  e.  RR* )  ->  (
( abs `  X
)  <_  R  <->  -.  R  <  ( abs `  X
) ) )
6822, 7, 67syl2anc 642 . 2  |-  ( ph  ->  ( ( abs `  X
)  <_  R  <->  -.  R  <  ( abs `  X
) ) )
6966, 68mpbird 223 1  |-  ( ph  ->  ( abs `  X
)  <_  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884    / cdiv 9439   2c2 9811   NN0cn0 9981   [,]cicc 10675    seq cseq 11062   ^cexp 11120   abscabs 11735    ~~> cli 11974
This theorem is referenced by:  pserdvlem2  19820  abelthlem1  19823  logtayl  20023
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  ax-addf 8832  ax-mulf 8833
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-int 3879  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-se 4369  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-isom 5280  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-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  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-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175
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