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Theorem radcnvlem2 21822
Description: Lemma for radcnvlt1 21826, radcnvle 21828. 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 10891 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 10591 . . 3  |-  1  e.  NN0
32a1i 11 . 2  |-  ( ph  ->  1  e.  NN0 )
4 id 22 . . . . . 6  |-  ( m  =  k  ->  m  =  k )
5 fveq2 5688 . . . . . . 7  |-  ( m  =  k  ->  (
( G `  X
) `  m )  =  ( ( G `
 X ) `  k ) )
65fveq2d 5692 . . . . . 6  |-  ( m  =  k  ->  ( abs `  ( ( G `
 X ) `  m ) )  =  ( abs `  (
( G `  X
) `  k )
) )
74, 6oveq12d 6108 . . . . 5  |-  ( m  =  k  ->  (
m  x.  ( abs `  ( ( G `  X ) `  m
) ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
8 eqid 2441 . . . . 5  |-  ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) )  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
 X ) `  m ) ) ) )
9 ovex 6115 . . . . 5  |-  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  _V
107, 8, 9fvmpt 5771 . . . 4  |-  ( k  e.  NN0  ->  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  =  ( k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
1110adantl 463 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  =  ( k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
12 nn0re 10584 . . . . 5  |-  ( k  e.  NN0  ->  k  e.  RR )
1312adantl 463 . . . 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 21820 . . . . . 6  |-  ( ph  ->  ( G `  X
) : NN0 --> CC )
1817ffvelrnda 5840 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  X ) `  k )  e.  CC )
1918abscld 12918 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  RR )
2013, 19remulcld 9410 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  RR )
2111, 20eqeltrd 2515 . 2  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  e.  RR )
22 fvco3 5765 . . . 4  |-  ( ( ( G `  X
) : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( abs  o.  ( G `  X ) ) `  k )  =  ( abs `  (
( G `  X
) `  k )
) )
2317, 22sylan 468 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( abs  o.  ( G `  X ) ) `  k )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
2419recnd 9408 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  CC )
2523, 24eqeltrd 2515 . 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 4380 . . 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 21821 . 2  |-  ( ph  ->  seq 0 (  +  ,  ( m  e. 
NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) )  e.  dom  ~~>  )
31 1red 9397 . 2  |-  ( ph  ->  1  e.  RR )
32 1red 9397 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  1  e.  RR )
33 elnnuz 10893 . . . . . 6  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
34 nnnn0 10582 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
3533, 34sylbir 213 . . . . 5  |-  ( k  e.  ( ZZ>= `  1
)  ->  k  e.  NN0 )
3635, 13sylan2 471 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  k  e.  RR )
3735, 19sylan2 471 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  RR )
3818absge0d 12926 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  ( abs `  ( ( G `  X ) `
 k ) ) )
3935, 38sylan2 471 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  0  <_  ( abs `  ( ( G `  X ) `
 k ) ) )
40 eluzle 10869 . . . . 5  |-  ( k  e.  ( ZZ>= `  1
)  ->  1  <_  k )
4140adantl 463 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  1  <_  k )
4232, 36, 37, 39, 41lemul1ad 10268 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) )  <_  (
k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
43 absidm 12807 . . . . . 6  |-  ( ( ( G `  X
) `  k )  e.  CC  ->  ( abs `  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4418, 43syl 16 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4523fveq2d 5692 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( abs `  ( abs `  (
( G `  X
) `  k )
) ) )
4624mulid2d 9400 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4744, 45, 463eqtr4d 2483 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) ) )
4835, 47sylan2 471 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) ) )
4911oveq2d 6106 . . . . 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
) ) ) ) )
5020recnd 9408 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  CC )
5150mulid2d 9400 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( k  x.  ( abs `  (
( G `  X
) `  k )
) ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5249, 51eqtrd 2473 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5335, 52sylan2 471 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5442, 48, 533brtr4d 4319 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  <_  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) ) )
551, 3, 21, 25, 30, 31, 54cvgcmpce 13277 1  |-  ( ph  ->  seq 0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   class class class wbr 4289    e. cmpt 4347   dom cdm 4836    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415   NNcn 10318   NN0cn0 10575   ZZ>=cuz 10857    seqcseq 11802   ^cexp 11861   abscabs 12719    ~~> cli 12958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ico 11302  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160
This theorem is referenced by:  radcnvlem3  21823  radcnvlt1  21826
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