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Theorem radcnvlt1 19810
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 8892 . . . . . . 7  |-  RR  C_  RR*
3 radcnvlt.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
43abscld 11934 . . . . . . 7  |-  ( ph  ->  ( abs `  X
)  e.  RR )
52, 4sseldi 3191 . . . . . 6  |-  ( ph  ->  ( abs `  X
)  e.  RR* )
6 iccssxr 10748 . . . . . . 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 19809 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
116, 10sseldi 3191 . . . . . 6  |-  ( ph  ->  R  e.  RR* )
12 xrltnle 8907 . . . . . 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 4046 . . . . . 6  |-  ( R  <_  ( abs `  X
)  <->  sup ( { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )  <_  ( abs `  X
) )
16 ssrab2 3271 . . . . . . . 8  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR
1716, 2sstri 3201 . . . . . . 7  |-  { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } 
C_  RR*
18 supxrleub 10661 . . . . . . 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 5541 . . . . . . . 8  |-  ( r  =  s  ->  ( G `  r )  =  ( G `  s ) )
2221seqeq3d 11070 . . . . . . 7  |-  ( r  =  s  ->  seq  0 (  +  , 
( G `  r
) )  =  seq  0 (  +  , 
( G `  s
) ) )
2322eleq1d 2362 . . . . . 6  |-  ( r  =  s  ->  (  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  <->  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  ) )
2423ralrab 2940 . . . . 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 2602 . . 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 8918 . . . . . . 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 8877 . . . . . . . 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 8854 . . . . . . . . . . . 12  |-  0  e.  RR
3837a1i 10 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  e.  RR )
3933abscld 11934 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  X )  e.  RR )
4033absge0d 11942 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  ( abs `  X ) )
4138, 39, 34, 40, 36lelttrd 8990 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <  s )
4238, 34, 41ltled 8983 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  0  <_  s )
4334, 42absidd 11921 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  RR )  /\  (  seq  0 (  +  , 
( G `  s
) )  e.  dom  ~~>  /\  ( abs `  X
)  <  s )
)  ->  ( abs `  s )  =  s )
4436, 43breqtrrd 4065 . . . . . . . 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 19805 . . . . . . 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 19806 . . . . . . 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 2680 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   dom cdm 4705    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884   NN0cn0 9981   [,]cicc 10675    seq cseq 11062   ^cexp 11120   abscabs 11735    ~~> cli 11974
This theorem is referenced by:  radcnvlt2  19811  dvradcnv  19813  pserulm  19814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175
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