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Theorem radcnvlem2 20330
Description: Lemma for radcnvlt1 20334, radcnvle 20336. 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 10520 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 10237 . . 3  |-  1  e.  NN0
32a1i 11 . 2  |-  ( ph  ->  1  e.  NN0 )
4 id 20 . . . . . 6  |-  ( m  =  k  ->  m  =  k )
5 fveq2 5728 . . . . . . 7  |-  ( m  =  k  ->  (
( G `  X
) `  m )  =  ( ( G `
 X ) `  k ) )
65fveq2d 5732 . . . . . 6  |-  ( m  =  k  ->  ( abs `  ( ( G `
 X ) `  m ) )  =  ( abs `  (
( G `  X
) `  k )
) )
74, 6oveq12d 6099 . . . . 5  |-  ( m  =  k  ->  (
m  x.  ( abs `  ( ( G `  X ) `  m
) ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
8 eqid 2436 . . . . 5  |-  ( m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) )  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
 X ) `  m ) ) ) )
9 ovex 6106 . . . . 5  |-  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  _V
107, 8, 9fvmpt 5806 . . . 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 10230 . . . . 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 20328 . . . . . 6  |-  ( ph  ->  ( G `  X
) : NN0 --> CC )
1817ffvelrnda 5870 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( G `  X ) `  k )  e.  CC )
1918abscld 12238 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  RR )
2013, 19remulcld 9116 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  RR )
2111, 20eqeltrd 2510 . 2  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
m  e.  NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) `  k )  e.  RR )
22 fvco3 5800 . . . 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 9114 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( G `  X ) `  k
) )  e.  CC )
2523, 24eqeltrd 2510 . 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 4300 . . 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 20329 . 2  |-  ( ph  ->  seq  0 (  +  ,  ( m  e. 
NN0  |->  ( m  x.  ( abs `  (
( G `  X
) `  m )
) ) ) )  e.  dom  ~~>  )
31 1re 9090 . . 3  |-  1  e.  RR
3231a1i 11 . 2  |-  ( ph  ->  1  e.  RR )
3331a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  1  e.  RR )
34 elnnuz 10522 . . . . . 6  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
35 nnnn0 10228 . . . . . 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 12246 . . . . 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 10498 . . . . 5  |-  ( k  e.  ( ZZ>= `  1
)  ->  1  <_  k )
4241adantl 453 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  1  <_  k )
4333, 37, 38, 40, 42lemul1ad 9950 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  1 )
)  ->  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) )  <_  (
k  x.  ( abs `  ( ( G `  X ) `  k
) ) ) )
44 absidm 12127 . . . . . 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 5732 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( abs `  ( ( abs  o.  ( G `  X ) ) `  k ) )  =  ( abs `  ( abs `  (
( G `  X
) `  k )
) ) )
4724mulid2d 9106 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( abs `  (
( G `  X
) `  k )
) )  =  ( abs `  ( ( G `  X ) `
 k ) ) )
4845, 46, 473eqtr4d 2478 . . . 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 6097 . . . . 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 9114 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  x.  ( abs `  (
( G `  X
) `  k )
) )  e.  CC )
5251mulid2d 9106 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( 1  x.  ( k  x.  ( abs `  (
( G `  X
) `  k )
) ) )  =  ( k  x.  ( abs `  ( ( G `
 X ) `  k ) ) ) )
5350, 52eqtrd 2468 . . . 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 4242 . 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 12597 1  |-  ( ph  ->  seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212    e. cmpt 4266   dom cdm 4878    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121   NNcn 10000   NN0cn0 10221   ZZ>=cuz 10488    seq cseq 11323   ^cexp 11382   abscabs 12039    ~~> cli 12278
This theorem is referenced by:  radcnvlem3  20331  radcnvlt1  20334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480
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