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Theorem radcnvlt1 21883
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 9427 . . . . . . 7  |-  RR  C_  RR*
3 radcnvlt.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
43abscld 12922 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
52, 4sseldi 3354 . . . . . 6  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
6 iccssxr 11378 . . . . . . 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 21882 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
116, 10sseldi 3354 . . . . . 6  |-  ( ph  ->  R  e.  RR* )
12 xrltnle 9443 . . . . . 6  |-  ( ( ( abs `  X
)  e.  RR*  /\  R  e.  RR* )  ->  (
( abs `  X
)  <  R  <->  -.  R  <_  ( abs `  X
) ) )
135, 11, 12syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( abs `  X
)  <  R  <->  -.  R  <_  ( abs `  X
) ) )
141, 13mpbid 210 . . . 4  |-  ( ph  ->  -.  R  <_  ( abs `  X ) )
159breq1i 4299 . . . . . 6  |-  ( R  <_  ( abs `  X
)  <->  sup ( { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )  <_  ( abs `  X
) )
16 ssrab2 3437 . . . . . . . 8  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
1716, 2sstri 3365 . . . . . . 7  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
18 supxrleub 11289 . . . . . . 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 663 . . . . . 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 257 . . . . 5  |-  ( ph  ->  ( R  <_  ( abs `  X )  <->  A. s  e.  { r  e.  RR  |  seq 0 (  +  ,  ( G `  r ) )  e. 
dom 
~~>  } s  <_  ( abs `  X ) ) )
21 fveq2 5691 . . . . . . . 8  |-  ( r  =  s  ->  ( G `  r )  =  ( G `  s ) )
2221seqeq3d 11814 . . . . . . 7  |-  ( r  =  s  ->  seq 0 (  +  , 
( G `  r
) )  =  seq 0 (  +  , 
( G `  s
) ) )
2322eleq1d 2509 . . . . . 6  |-  ( r  =  s  ->  (  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  ) )
2423ralrab 3121 . . . . 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 261 . . . 4  |-  ( ph  ->  ( R  <_  ( abs `  X )  <->  A. s  e.  RR  (  seq 0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  ->  s  <_  ( abs `  X
) ) ) )
2614, 25mtbid 300 . . 3  |-  ( ph  ->  -.  A. s  e.  RR  (  seq 0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  ->  s  <_  ( abs `  X
) ) )
27 rexanali 2761 . . 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 212 . 2  |-  ( ph  ->  E. s  e.  RR  (  seq 0 (  +  ,  ( G `  s ) )  e. 
dom 
~~>  /\  -.  s  <_ 
( abs `  X
) ) )
29 ltnle 9454 . . . . . . 7  |-  ( ( ( abs `  X
)  e.  RR  /\  s  e.  RR )  ->  ( ( abs `  X
)  <  s  <->  -.  s  <_  ( abs `  X
) ) )
304, 29sylan 471 . . . . . 6  |-  ( (
ph  /\  s  e.  RR )  ->  ( ( abs `  X )  <  s  <->  -.  s  <_  ( abs `  X
) ) )
3130adantr 465 . . . . 5  |-  ( ( ( ph  /\  s  e.  RR )  /\  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  )  ->  ( ( abs `  X )  <  s  <->  -.  s  <_  ( abs `  X ) ) )
328ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  A : NN0
--> CC )
333ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  X  e.  CC )
34 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  s  e.  RR )
3534recnd 9412 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  s  e.  CC )
36 simprr 756 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  <  s
)
37 0red 9387 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  e.  RR )
3833abscld 12922 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  e.  RR )
3933absge0d 12930 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  ( abs `  X ) )
4037, 38, 34, 39, 36lelttrd 9529 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <  s )
4137, 34, 40ltled 9522 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  s )
4234, 41absidd 12909 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  s )  =  s )
4336, 42breqtrrd 4318 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  <  ( abs `  s ) )
44 simprl 755 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  seq 0
(  +  ,  ( G `  s ) )  e.  dom  ~~>  )
45 radcnvlt1.h . . . . . . . 8  |-  H  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
 X ) `  m ) ) ) )
467, 32, 33, 35, 43, 44, 45radcnvlem1 21878 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  seq 0
(  +  ,  H
)  e.  dom  ~~>  )
477, 32, 33, 35, 43, 44radcnvlem2 21879 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  seq 0
(  +  ,  ( abs  o.  ( G `
 X ) ) )  e.  dom  ~~>  )
4846, 47jca 532 . . . . . 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  ~~>  ) )
4948expr 615 . . . . 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  ~~>  ) ) )
5031, 49sylbird 235 . . . 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  ~~>  ) ) )
5150expimpd 603 . . 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  ~~>  ) ) )
5251rexlimdva 2841 . 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  ~~>  ) ) )
5328, 52mpd 15 1  |-  ( ph  ->  (  seq 0 (  +  ,  H )  e.  dom  ~~>  /\  seq 0 (  +  , 
( abs  o.  ( G `  X )
) )  e.  dom  ~~>  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   {crab 2719    C_ wss 3328   class class class wbr 4292    e. cmpt 4350   dom cdm 4840    o. ccom 4844   -->wf 5414   ` cfv 5418  (class class class)co 6091   supcsup 7690   CCcc 9280   RRcr 9281   0cc0 9282    + caddc 9285    x. cmul 9287   +oocpnf 9415   RR*cxr 9417    < clt 9418    <_ cle 9419   NN0cn0 10579   [,]cicc 11303    seqcseq 11806   ^cexp 11865   abscabs 12723    ~~> cli 12962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164
This theorem is referenced by:  radcnvlt2  21884  dvradcnv  21886  pserulm  21887
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