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Theorem radcnvlem2 20283
Description: Lemma for radcnvlt1 20287, radcnvle 20289. If  X is a point closer to zero than  Y and the power series converges at 
Y, then it converges absolutely at  X. (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 )
psergf.x  |-  ( ph  ->  X  e.  CC )
radcnvlem2.y  |-  ( ph  ->  Y  e.  CC )
radcnvlem2.a  |-  ( ph  ->  ( abs `  X
)  <  ( abs `  Y ) )
radcnvlem2.c  |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e. 
dom 
~~>  )
Assertion
Ref Expression
radcnvlem2  |-  ( ph  ->  seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  )
Distinct variable group:    x, n, A
Allowed substitution hints:    ph( x, n)    G( x, n)    X( x, n)    Y( x, n)

Proof of Theorem radcnvlem2
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 10476 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 10193 . . 3  |-  1  e.  NN0
32a1i 11 . 2  |-  ( ph  ->  1  e.  NN0 )
4 id 20 . . . . . 6  |-  ( m  =  k  ->  m  =  k )
5 fveq2 5687 . . . . . . 7  |-  ( m  =  k  ->  (
( G `  X
) `  m )  =  ( ( G `
 X ) `  k ) )
65fveq2d 5691 . . . . . 6  |-  ( m  =  k  ->  ( abs `  ( ( G `
 X ) `  m ) )  =  ( abs `  (
( G `  X
) `  k )
) )
74, 6oveq12d 6058 . . . . 5  |-  ( m  =  k  ->  (
m  x.  ( abs `  ( ( G `  X ) `  m
) ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
8 eqid 2404 . . . . 5  |-  ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) )  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
 X ) `  m ) ) ) )
9 ovex 6065 . . . . 5  |-  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  _V
107, 8, 9fvmpt 5765 . . . 4  |-  ( k  e.  NN0  ->  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  =  ( k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
1110adantl 453 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  =  ( k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
12 nn0re 10186 . . . . 5  |-  ( k  e.  NN0  ->  k  e.  RR )
1312adantl 453 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  RR )
14 pser.g . . . . . . 7  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
15 radcnv.a . . . . . . 7  |-  ( ph  ->  A : NN0 --> CC )
16 psergf.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
1714, 15, 16psergf 20281 . . . . . 6  |-  ( ph  ->  ( G `  X
) : NN0 --> CC )
1817ffvelrnda 5829 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  X ) `  k )  e.  CC )
1918abscld 12193 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  RR )
2013, 19remulcld 9072 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  RR )
2111, 20eqeltrd 2478 . 2  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  e.  RR )
22 fvco3 5759 . . . 4  |-  ( ( ( G `  X
) : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( abs  o.  ( G `  X ) ) `  k )  =  ( abs `  (
( G `  X
) `  k )
) )
2317, 22sylan 458 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( abs  o.  ( G `  X ) ) `  k )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
2419recnd 9070 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  CC )
2523, 24eqeltrd 2478 . 2  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( abs  o.  ( G `  X ) ) `  k )  e.  CC )
26 radcnvlem2.y . . 3  |-  ( ph  ->  Y  e.  CC )
27 radcnvlem2.a . . 3  |-  ( ph  ->  ( abs `  X
)  <  ( abs `  Y ) )
28 radcnvlem2.c . . 3  |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e. 
dom 
~~>  )
297cbvmptv 4260 . . 3  |-  ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) )  =  ( k  e.  NN0  |->  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
3014, 15, 16, 26, 27, 28, 29radcnvlem1 20282 . 2  |-  ( ph  ->  seq  0 (  +  ,  ( m  e. 
NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) )  e.  dom  ~~>  )
31 1re 9046 . . 3  |-  1  e.  RR
3231a1i 11 . 2  |-  ( ph  ->  1  e.  RR )
3331a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  1  e.  RR )
34 elnnuz 10478 . . . . . 6  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
35 nnnn0 10184 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
3634, 35sylbir 205 . . . . 5  |-  ( k  e.  ( ZZ>= `  1
)  ->  k  e.  NN0 )
3736, 13sylan2 461 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  k  e.  RR )
3836, 19sylan2 461 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  RR )
3918absge0d 12201 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  ( abs `  ( ( G `  X ) `
 k ) ) )
4036, 39sylan2 461 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  0  <_  ( abs `  ( ( G `  X ) `
 k ) ) )
41 eluzle 10454 . . . . 5  |-  ( k  e.  ( ZZ>= `  1
)  ->  1  <_  k )
4241adantl 453 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  1  <_  k )
4333, 37, 38, 40, 42lemul1ad 9906 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) )  <_  (
k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
44 absidm 12082 . . . . . 6  |-  ( ( ( G `  X
) `  k )  e.  CC  ->  ( abs `  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4518, 44syl 16 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4623fveq2d 5691 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( abs `  ( abs `  (
( G `  X
) `  k )
) ) )
4724mulid2d 9062 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4845, 46, 473eqtr4d 2446 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) ) )
4936, 48sylan2 461 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) ) )
5011oveq2d 6056 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) )  =  ( 1  x.  (
k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) ) )
5120recnd 9070 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  CC )
5251mulid2d 9062 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( k  x.  ( abs `  (
( G `  X
) `  k )
) ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5350, 52eqtrd 2436 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5436, 53sylan2 461 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5543, 49, 543brtr4d 4202 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  <_  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) ) )
561, 3, 21, 25, 30, 32, 55cvgcmpce 12552 1  |-  ( ph  ->  seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172    e. cmpt 4226   dom cdm 4837    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077   NNcn 9956   NN0cn0 10177   ZZ>=cuz 10444    seq cseq 11278   ^cexp 11337   abscabs 11994    ~~> cli 12233
This theorem is referenced by:  radcnvlem3  20284  radcnvlt1  20287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435
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