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Theorem radcnvlt1 19794
Description: If  X is within the open disk of radius  R centered at zero, then the infinite series converges absolutely at  X, and also converges when the series is multiplied by  n. (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* ,  <  )
radcnvlt.x  |-  ( ph  ->  X  e.  CC )
radcnvlt.a  |-  ( ph  ->  ( abs `  X
)  <  R )
radcnvlt1.h  |-  H  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
 X ) `  m ) ) ) )
Assertion
Ref Expression
radcnvlt1  |-  ( ph  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\  seq  0 (  +  , 
( abs  o.  ( G `  X )
) )  e.  dom  ~~>  ) )
Distinct variable groups:    m, n, x, A    m, H    ph, m    m, X    m, r, G
Allowed substitution hints:    ph( x, n, r)    A( r)    R( x, m, n, r)    G( x, n)    H( x, n, r)    X( x, n, r)

Proof of Theorem radcnvlt1
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 radcnvlt.a . . . . 5  |-  ( ph  ->  ( abs `  X
)  <  R )
2 ressxr 8876 . . . . . . 7  |-  RR  C_  RR*
3 radcnvlt.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
43abscld 11918 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
52, 4sseldi 3178 . . . . . 6  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
6 iccssxr 10732 . . . . . . 7  |-  ( 0 [,]  +oo )  C_  RR*
7 pser.g . . . . . . . 8  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
8 radcnv.a . . . . . . . 8  |-  ( ph  ->  A : NN0 --> CC )
9 radcnv.r . . . . . . . 8  |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
107, 8, 9radcnvcl 19793 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
116, 10sseldi 3178 . . . . . 6  |-  ( ph  ->  R  e.  RR* )
12 xrltnle 8891 . . . . . 6  |-  ( ( ( abs `  X
)  e.  RR*  /\  R  e.  RR* )  ->  (
( abs `  X
)  <  R  <->  -.  R  <_  ( abs `  X
) ) )
135, 11, 12syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( abs `  X
)  <  R  <->  -.  R  <_  ( abs `  X
) ) )
141, 13mpbid 201 . . . 4  |-  ( ph  ->  -.  R  <_  ( abs `  X ) )
159breq1i 4030 . . . . . 6  |-  ( R  <_  ( abs `  X
)  <->  sup ( { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )  <_  ( abs `  X
) )
16 ssrab2 3258 . . . . . . . 8  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
1716, 2sstri 3188 . . . . . . 7  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
18 supxrleub 10645 . . . . . . 7  |-  ( ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  }  C_  RR*  /\  ( abs `  X )  e. 
RR* )  ->  ( sup ( { r  e.  RR  |  seq  0
(  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )  <_ 
( abs `  X
)  <->  A. s  e.  {
r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } s  <_  ( abs `  X ) ) )
1917, 5, 18sylancr 644 . . . . . 6  |-  ( ph  ->  ( sup ( { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )  <_  ( abs `  X
)  <->  A. s  e.  {
r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } s  <_  ( abs `  X ) ) )
2015, 19syl5bb 248 . . . . 5  |-  ( ph  ->  ( R  <_  ( abs `  X )  <->  A. s  e.  { r  e.  RR  |  seq  0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  } s  <_  ( abs `  X ) ) )
21 fveq2 5525 . . . . . . . 8  |-  ( r  =  s  ->  ( G `  r )  =  ( G `  s ) )
2221seqeq3d 11054 . . . . . . 7  |-  ( r  =  s  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  s
) ) )
2322eleq1d 2349 . . . . . 6  |-  ( r  =  s  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  ) )
2423ralrab 2927 . . . . 5  |-  ( A. s  e.  { r  e.  RR  |  seq  0
(  +  ,  ( G `  r ) )  e.  dom  ~~>  } s  <_  ( abs `  X
)  <->  A. s  e.  RR  (  seq  0 (  +  ,  ( G `  s ) )  e. 
dom 
~~>  ->  s  <_  ( abs `  X ) ) )
2520, 24syl6bb 252 . . . 4  |-  ( ph  ->  ( R  <_  ( abs `  X )  <->  A. s  e.  RR  (  seq  0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  ->  s  <_  ( abs `  X
) ) ) )
2614, 25mtbid 291 . . 3  |-  ( ph  ->  -.  A. s  e.  RR  (  seq  0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  ->  s  <_  ( abs `  X
) ) )
27 rexanali 2589 . . 3  |-  ( E. s  e.  RR  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\ 
-.  s  <_  ( abs `  X ) )  <->  -.  A. s  e.  RR  (  seq  0 (  +  ,  ( G `  s ) )  e. 
dom 
~~>  ->  s  <_  ( abs `  X ) ) )
2826, 27sylibr 203 . 2  |-  ( ph  ->  E. s  e.  RR  (  seq  0 (  +  ,  ( G `  s ) )  e. 
dom 
~~>  /\  -.  s  <_ 
( abs `  X
) ) )
29 ltnle 8902 . . . . . . 7  |-  ( ( ( abs `  X
)  e.  RR  /\  s  e.  RR )  ->  ( ( abs `  X
)  <  s  <->  -.  s  <_  ( abs `  X
) ) )
304, 29sylan 457 . . . . . 6  |-  ( (
ph  /\  s  e.  RR )  ->  ( ( abs `  X )  <  s  <->  -.  s  <_  ( abs `  X
) ) )
3130adantr 451 . . . . 5  |-  ( ( ( ph  /\  s  e.  RR )  /\  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  )  ->  ( ( abs `  X )  <  s  <->  -.  s  <_  ( abs `  X ) ) )
328ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  A : NN0
--> CC )
333ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  X  e.  CC )
34 simplr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  s  e.  RR )
3534recnd 8861 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  s  e.  CC )
36 simprr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  <  s
)
37 0re 8838 . . . . . . . . . . . 12  |-  0  e.  RR
3837a1i 10 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  e.  RR )
3933abscld 11918 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  e.  RR )
4033absge0d 11926 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  ( abs `  X ) )
4138, 39, 34, 40, 36lelttrd 8974 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <  s )
4238, 34, 41ltled 8967 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  s )
4334, 42absidd 11905 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  s )  =  s )
4436, 43breqtrrd 4049 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  <  ( abs `  s ) )
45 simprl 732 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  seq  0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  )
46 radcnvlt1.h . . . . . . . 8  |-  H  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
 X ) `  m ) ) ) )
477, 32, 33, 35, 44, 45, 46radcnvlem1 19789 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  seq  0
(  +  ,  H
)  e.  dom  ~~>  )
487, 32, 33, 35, 44, 45radcnvlem2 19790 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  seq  0
(  +  ,  ( abs  o.  ( G `
 X ) ) )  e.  dom  ~~>  )
4947, 48jca 518 . . . . . 6  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\ 
seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  ) )
5049expr 598 . . . . 5  |-  ( ( ( ph  /\  s  e.  RR )  /\  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  )  ->  ( ( abs `  X )  <  s  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\  seq  0 (  +  , 
( abs  o.  ( G `  X )
) )  e.  dom  ~~>  ) ) )
5131, 50sylbird 226 . . . 4  |-  ( ( ( ph  /\  s  e.  RR )  /\  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  )  ->  ( -.  s  <_  ( abs `  X
)  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\ 
seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  ) ) )
5251expimpd 586 . . 3  |-  ( (
ph  /\  s  e.  RR )  ->  ( (  seq  0 (  +  ,  ( G `  s ) )  e. 
dom 
~~>  /\  -.  s  <_ 
( abs `  X
) )  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\ 
seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  ) ) )
5352rexlimdva 2667 . 2  |-  ( ph  ->  ( E. s  e.  RR  (  seq  0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  /\  -.  s  <_  ( abs `  X
) )  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\ 
seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  ) ) )
5428, 53mpd 14 1  |-  ( ph  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\  seq  0 (  +  , 
( abs  o.  ( G `  X )
) )  e.  dom  ~~>  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   NN0cn0 9965   [,]cicc 10659    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  radcnvlt2  19795  dvradcnv  19797  pserulm  19798
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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