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Theorem radcnvlem2 19843
Description: Lemma for radcnvlt1 19847, radcnvle 19849. 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 10309 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 10028 . . 3  |-  1  e.  NN0
32a1i 10 . 2  |-  ( ph  ->  1  e.  NN0 )
4 id 19 . . . . . 6  |-  ( m  =  k  ->  m  =  k )
5 fveq2 5563 . . . . . . 7  |-  ( m  =  k  ->  (
( G `  X
) `  m )  =  ( ( G `
 X ) `  k ) )
65fveq2d 5567 . . . . . 6  |-  ( m  =  k  ->  ( abs `  ( ( G `
 X ) `  m ) )  =  ( abs `  (
( G `  X
) `  k )
) )
74, 6oveq12d 5918 . . . . 5  |-  ( m  =  k  ->  (
m  x.  ( abs `  ( ( G `  X ) `  m
) ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
8 eqid 2316 . . . . 5  |-  ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) )  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
 X ) `  m ) ) ) )
9 ovex 5925 . . . . 5  |-  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  _V
107, 8, 9fvmpt 5640 . . . 4  |-  ( k  e.  NN0  ->  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  =  ( k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
1110adantl 452 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  =  ( k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
12 nn0re 10021 . . . . 5  |-  ( k  e.  NN0  ->  k  e.  RR )
1312adantl 452 . . . 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 19841 . . . . . 6  |-  ( ph  ->  ( G `  X
) : NN0 --> CC )
18 ffvelrn 5701 . . . . . 6  |-  ( ( ( G `  X
) : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( G `  X ) `  k
)  e.  CC )
1917, 18sylan 457 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  X ) `  k )  e.  CC )
2019abscld 11965 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  RR )
2113, 20remulcld 8908 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  RR )
2211, 21eqeltrd 2390 . 2  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  e.  RR )
23 fvco3 5634 . . . 4  |-  ( ( ( G `  X
) : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( abs  o.  ( G `  X ) ) `  k )  =  ( abs `  (
( G `  X
) `  k )
) )
2417, 23sylan 457 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( abs  o.  ( G `  X ) ) `  k )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
2520recnd 8906 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  CC )
2624, 25eqeltrd 2390 . 2  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( abs  o.  ( G `  X ) ) `  k )  e.  CC )
27 radcnvlem2.y . . 3  |-  ( ph  ->  Y  e.  CC )
28 radcnvlem2.a . . 3  |-  ( ph  ->  ( abs `  X
)  <  ( abs `  Y ) )
29 radcnvlem2.c . . 3  |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e. 
dom 
~~>  )
307cbvmptv 4148 . . 3  |-  ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) )  =  ( k  e.  NN0  |->  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
3114, 15, 16, 27, 28, 29, 30radcnvlem1 19842 . 2  |-  ( ph  ->  seq  0 (  +  ,  ( m  e. 
NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) )  e.  dom  ~~>  )
32 1re 8882 . . 3  |-  1  e.  RR
3332a1i 10 . 2  |-  ( ph  ->  1  e.  RR )
3432a1i 10 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  1  e.  RR )
35 elnnuz 10311 . . . . . 6  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
36 nnnn0 10019 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
3735, 36sylbir 204 . . . . 5  |-  ( k  e.  ( ZZ>= `  1
)  ->  k  e.  NN0 )
3837, 13sylan2 460 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  k  e.  RR )
3937, 20sylan2 460 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  RR )
4019absge0d 11973 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  ( abs `  ( ( G `  X ) `
 k ) ) )
4137, 40sylan2 460 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  0  <_  ( abs `  ( ( G `  X ) `
 k ) ) )
42 eluzle 10287 . . . . 5  |-  ( k  e.  ( ZZ>= `  1
)  ->  1  <_  k )
4342adantl 452 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  1  <_  k )
4434, 38, 39, 41, 43lemul1ad 9741 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) )  <_  (
k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
45 absidm 11854 . . . . . 6  |-  ( ( ( G `  X
) `  k )  e.  CC  ->  ( abs `  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4619, 45syl 15 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4724fveq2d 5567 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( abs `  ( abs `  (
( G `  X
) `  k )
) ) )
4825mulid2d 8898 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4946, 47, 483eqtr4d 2358 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) ) )
5037, 49sylan2 460 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) ) )
5111oveq2d 5916 . . . . 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
) ) ) ) )
5221recnd 8906 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  CC )
5352mulid2d 8898 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( k  x.  ( abs `  (
( G `  X
) `  k )
) ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5451, 53eqtrd 2348 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5537, 54sylan2 460 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5644, 50, 553brtr4d 4090 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  <_  ( 1  x.  ( ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k ) ) )
571, 3, 22, 26, 31, 33, 56cvgcmpce 12323 1  |-  ( ph  ->  seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   class class class wbr 4060    e. cmpt 4114   dom cdm 4726    o. ccom 4730   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    < clt 8912    <_ cle 8913   NNcn 9791   NN0cn0 10012   ZZ>=cuz 10277    seq cseq 11093   ^cexp 11151   abscabs 11766    ~~> cli 12005
This theorem is referenced by:  radcnvlem3  19844  radcnvlt1  19847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-pm 6818  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-ico 10709  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-sum 12206
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