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Theorem radcnvlt1 22938
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 9654 . . . . . . 7  |-  RR  C_  RR*
3 radcnvlt.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
43abscld 13278 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
52, 4sseldi 3497 . . . . . 6  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
6 iccssxr 11632 . . . . . . 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 22937 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
116, 10sseldi 3497 . . . . . 6  |-  ( ph  ->  R  e.  RR* )
12 xrltnle 9670 . . . . . 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 4463 . . . . . 6  |-  ( R  <_  ( abs `  X
)  <->  sup ( { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )  <_  ( abs `  X
) )
16 ssrab2 3581 . . . . . . . 8  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
1716, 2sstri 3508 . . . . . . 7  |-  { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
18 supxrleub 11543 . . . . . . 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 5872 . . . . . . . 8  |-  ( r  =  s  ->  ( G `  r )  =  ( G `  s ) )
2221seqeq3d 12117 . . . . . . 7  |-  ( r  =  s  ->  seq 0 (  +  , 
( G `  r
) )  =  seq 0 (  +  , 
( G `  s
) ) )
2322eleq1d 2526 . . . . . 6  |-  ( r  =  s  ->  (  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  ) )
2423ralrab 3261 . . . . 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 2910 . . 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 9681 . . . . . . 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 755 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  s  e.  RR )
3534recnd 9639 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  s  e.  CC )
36 simprr 757 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  <  s
)
37 0red 9614 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  e.  RR )
3833abscld 13278 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  e.  RR )
3933absge0d 13286 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  ( abs `  X ) )
4037, 38, 34, 39, 36lelttrd 9757 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <  s )
4137, 34, 40ltled 9750 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  s )
4234, 41absidd 13265 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  s )  =  s )
4336, 42breqtrrd 4482 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq 0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  <  ( abs `  s ) )
44 simprl 756 . . . . . . . 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 22933 . . . . . . 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 22934 . . . . . . 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 2949 . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512    x. cmul 9514   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   NN0cn0 10816   [,]cicc 11557    seqcseq 12109   ^cexp 12168   abscabs 13078    ~~> cli 13318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-limsup 13305  df-clim 13322  df-rlim 13323  df-sum 13520
This theorem is referenced by:  radcnvlt2  22939  dvradcnv  22941  pserulm  22942
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