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Theorem radcnvlt1 20336
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 9131 . . . . . . 7  |-  RR  C_  RR*
3 radcnvlt.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
43abscld 12240 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
52, 4sseldi 3348 . . . . . 6  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
6 iccssxr 10995 . . . . . . 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 20335 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
116, 10sseldi 3348 . . . . . 6  |-  ( ph  ->  R  e.  RR* )
12 xrltnle 9146 . . . . . 6  |-  ( ( ( abs `  X
)  e.  RR*  /\  R  e.  RR* )  ->  (
( abs `  X
)  <  R  <->  -.  R  <_  ( abs `  X
) ) )
135, 11, 12syl2anc 644 . . . . 5  |-  ( ph  ->  ( ( abs `  X
)  <  R  <->  -.  R  <_  ( abs `  X
) ) )
141, 13mpbid 203 . . . 4  |-  ( ph  ->  -.  R  <_  ( abs `  X ) )
159breq1i 4221 . . . . . 6  |-  ( R  <_  ( abs `  X
)  <->  sup ( { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )  <_  ( abs `  X
) )
16 ssrab2 3430 . . . . . . . 8  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
1716, 2sstri 3359 . . . . . . 7  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
18 supxrleub 10907 . . . . . . 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 646 . . . . . 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 250 . . . . 5  |-  ( ph  ->  ( R  <_  ( abs `  X )  <->  A. s  e.  { r  e.  RR  |  seq  0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  } s  <_  ( abs `  X ) ) )
21 fveq2 5730 . . . . . . . 8  |-  ( r  =  s  ->  ( G `  r )  =  ( G `  s ) )
2221seqeq3d 11333 . . . . . . 7  |-  ( r  =  s  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  s
) ) )
2322eleq1d 2504 . . . . . 6  |-  ( r  =  s  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  ) )
2423ralrab 3098 . . . . 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 254 . . . 4  |-  ( ph  ->  ( R  <_  ( abs `  X )  <->  A. s  e.  RR  (  seq  0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  ->  s  <_  ( abs `  X
) ) ) )
2614, 25mtbid 293 . . 3  |-  ( ph  ->  -.  A. s  e.  RR  (  seq  0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  ->  s  <_  ( abs `  X
) ) )
27 rexanali 2753 . . 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 205 . 2  |-  ( ph  ->  E. s  e.  RR  (  seq  0 (  +  ,  ( G `  s ) )  e. 
dom 
~~>  /\  -.  s  <_ 
( abs `  X
) ) )
29 ltnle 9157 . . . . . . 7  |-  ( ( ( abs `  X
)  e.  RR  /\  s  e.  RR )  ->  ( ( abs `  X
)  <  s  <->  -.  s  <_  ( abs `  X
) ) )
304, 29sylan 459 . . . . . 6  |-  ( (
ph  /\  s  e.  RR )  ->  ( ( abs `  X )  <  s  <->  -.  s  <_  ( abs `  X
) ) )
3130adantr 453 . . . . 5  |-  ( ( ( ph  /\  s  e.  RR )  /\  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  )  ->  ( ( abs `  X )  <  s  <->  -.  s  <_  ( abs `  X ) ) )
328ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  A : NN0
--> CC )
333ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  X  e.  CC )
34 simplr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  s  e.  RR )
3534recnd 9116 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  s  e.  CC )
36 simprr 735 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  <  s
)
37 0re 9093 . . . . . . . . . . . 12  |-  0  e.  RR
3837a1i 11 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  e.  RR )
3933abscld 12240 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  e.  RR )
4033absge0d 12248 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  ( abs `  X ) )
4138, 39, 34, 40, 36lelttrd 9230 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <  s )
4238, 34, 41ltled 9223 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  s )
4334, 42absidd 12227 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  s )  =  s )
4436, 43breqtrrd 4240 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  <  ( abs `  s ) )
45 simprl 734 . . . . . . . 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 20331 . . . . . . 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 20332 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  seq  0
(  +  ,  ( abs  o.  ( G `
 X ) ) )  e.  dom  ~~>  )
4947, 48jca 520 . . . . . 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 600 . . . . 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 228 . . . 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 588 . . 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 2832 . 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 15 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   {crab 2711    C_ wss 3322   class class class wbr 4214    e. cmpt 4268   dom cdm 4880    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083   supcsup 7447   CCcc 8990   RRcr 8991   0cc0 8992    + caddc 8995    x. cmul 8997    +oocpnf 9119   RR*cxr 9121    < clt 9122    <_ cle 9123   NN0cn0 10223   [,]cicc 10921    seq cseq 11325   ^cexp 11384   abscabs 12041    ~~> cli 12280
This theorem is referenced by:  radcnvlt2  20337  dvradcnv  20339  pserulm  20340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482
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