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Theorem radcnvle 19796
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 10732 . . . . . . . 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 19793 . . . . . . . 8  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
72, 6sseldi 3178 . . . . . . 7  |-  ( ph  ->  R  e.  RR* )
87adantr 451 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  R  e.  RR* )
9 radcnvle.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
109abscld 11918 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
1110adantr 451 . . . . . 6  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR )
12 0xr 8878 . . . . . . . . . . 11  |-  0  e.  RR*
13 pnfxr 10455 . . . . . . . . . . 11  |-  +oo  e.  RR*
14 elicc1 10700 . . . . . . . . . . 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 10501 . . . . . . . 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 8876 . . . . . . . . . 10  |-  RR  C_  RR*
2221, 10sseldi 3178 . . . . . . . . 9  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
2322adantr 451 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  X )  e.  RR* )
24 xrltle 10483 . . . . . . . 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 10498 . . . . . 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 9949 . . . . 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 3258 . . . . . . 7  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
3332, 21sstri 3188 . . . . . 6  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
3428, 11readdcld 8862 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( R  +  ( abs `  X
) )  e.  RR )
3534rehalfcld 9958 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  RR )
364adantr 451 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  A : NN0
--> CC )
3735recnd 8861 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  e.  CC )
389adantr 451 . . . . . . . 8  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  X  e.  CC )
39 0re 8838 . . . . . . . . . . . 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 8974 . . . . . . . . . . 11  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <  ( ( R  +  ( abs `  X ) )  /  2 ) )
4340, 35, 42ltled 8967 . . . . . . . . . 10  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  0  <_  ( ( R  +  ( abs `  X ) )  /  2 ) )
4435, 43absidd 11905 . . . . . . . . 9  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( abs `  ( ( R  +  ( abs `  X ) )  /  2 ) )  =  ( ( R  +  ( abs `  X ) )  / 
2 ) )
45 avglt2 9950 . . . . . . . . . . 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 4043 . . . . . . . 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 19791 . . . . . . 7  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  seq  0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )  e.  dom  ~~>  )
52 fveq2 5525 . . . . . . . . . 10  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  ( G `  y )  =  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) )
5352seqeq3d 11054 . . . . . . . . 9  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  seq  0
(  +  ,  ( G `  y ) )  =  seq  0
(  +  ,  ( G `  ( ( R  +  ( abs `  X ) )  / 
2 ) ) ) )
5453eleq1d 2349 . . . . . . . 8  |-  ( y  =  ( ( R  +  ( abs `  X
) )  /  2
)  ->  (  seq  0 (  +  , 
( G `  y
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  (
( R  +  ( abs `  X ) )  /  2 ) ) )  e.  dom  ~~>  ) )
55 fveq2 5525 . . . . . . . . . . 11  |-  ( r  =  y  ->  ( G `  r )  =  ( G `  y ) )
5655seqeq3d 11054 . . . . . . . . . 10  |-  ( r  =  y  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  y
) ) )
5756eleq1d 2349 . . . . . . . . 9  |-  ( r  =  y  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  y
) )  e.  dom  ~~>  ) )
5857cbvrabv 2787 . . . . . . . 8  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  }  =  { y  e.  RR  |  seq  0
(  +  ,  ( G `  y ) )  e.  dom  ~~>  }
5954, 58elrab2 2925 . . . . . . 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 10643 . . . . . 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 4063 . . . 4  |-  ( (
ph  /\  R  <  ( abs `  X ) )  ->  ( ( R  +  ( abs `  X ) )  / 
2 )  <_  R
)
6435, 28lenltd 8965 . . . 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 8890 . . 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 1623    e. wcel 1684   {crab 2547    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868    / cdiv 9423   2c2 9795   NN0cn0 9965   [,]cicc 10659    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  pserdvlem2  19804  abelthlem1  19807  logtayl  20007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159
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