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Theorem psercnlem2 21864
Description: Lemma for psercn 21866. (Contributed by Mario Carneiro, 18-Mar-2015.)
Hypotheses
Ref Expression
pserf.g  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
pserf.f  |-  F  =  ( y  e.  S  |-> 
sum_ j  e.  NN0  ( ( G `  y ) `  j
) )
pserf.a  |-  ( ph  ->  A : NN0 --> CC )
pserf.r  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
psercn.s  |-  S  =  ( `' abs " (
0 [,) R ) )
psercnlem2.i  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
Assertion
Ref Expression
psercnlem2  |-  ( (
ph  /\  a  e.  S )  ->  (
a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) M )  /\  (
0 ( ball `  ( abs  o.  -  ) ) M )  C_  ( `' abs " ( 0 [,] M ) )  /\  ( `' abs " ( 0 [,] M
) )  C_  S
) )
Distinct variable groups:    j, a, n, r, x, y, A   
j, M, y    j, G, r, y    S, a, j, y    F, a    ph, a, j, y
Allowed substitution hints:    ph( x, n, r)    R( x, y, j, n, r, a)    S( x, n, r)    F( x, y, j, n, r)    G( x, n, a)    M( x, n, r, a)

Proof of Theorem psercnlem2
Dummy variables  k  w  z  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psercn.s . . . . . . 7  |-  S  =  ( `' abs " (
0 [,) R ) )
2 cnvimass 5184 . . . . . . . 8  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
3 absf 12817 . . . . . . . . 9  |-  abs : CC
--> RR
43fdmi 5559 . . . . . . . 8  |-  dom  abs  =  CC
52, 4sseqtri 3383 . . . . . . 7  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
61, 5eqsstri 3381 . . . . . 6  |-  S  C_  CC
76a1i 11 . . . . 5  |-  ( ph  ->  S  C_  CC )
87sselda 3351 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
98abscld 12914 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
108absge0d 12922 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
11 psercnlem2.i . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
1211simp2d 1001 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
13 0re 9378 . . . . . 6  |-  0  e.  RR
1411simp1d 1000 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
1514rpxrd 11020 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR* )
16 elico2 11351 . . . . . 6  |-  ( ( 0  e.  RR  /\  M  e.  RR* )  -> 
( ( abs `  a
)  e.  ( 0 [,) M )  <->  ( ( abs `  a )  e.  RR  /\  0  <_ 
( abs `  a
)  /\  ( abs `  a )  <  M
) ) )
1713, 15, 16sylancr 663 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  ( 0 [,) M )  <->  ( ( abs `  a )  e.  RR  /\  0  <_ 
( abs `  a
)  /\  ( abs `  a )  <  M
) ) )
189, 10, 12, 17mpbir3and 1171 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  ( 0 [,) M
) )
19 ffn 5554 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
20 elpreima 5818 . . . . 5  |-  ( abs 
Fn  CC  ->  ( a  e.  ( `' abs " ( 0 [,) M
) )  <->  ( a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) M ) ) ) )
213, 19, 20mp2b 10 . . . 4  |-  ( a  e.  ( `' abs " ( 0 [,) M
) )  <->  ( a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) M ) ) )
228, 18, 21sylanbrc 664 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( `' abs " (
0 [,) M ) ) )
23 eqid 2438 . . . . 5  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2423cnbl0 20328 . . . 4  |-  ( M  e.  RR*  ->  ( `' abs " ( 0 [,) M ) )  =  ( 0 (
ball `  ( abs  o. 
-  ) ) M ) )
2515, 24syl 16 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,) M ) )  =  ( 0 (
ball `  ( abs  o. 
-  ) ) M ) )
2622, 25eleqtrd 2514 . 2  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) M ) )
27 df-ico 11298 . . . . 5  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
28 df-icc 11299 . . . . 5  |-  [,]  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <_  v ) } )
29 idd 24 . . . . 5  |-  ( ( 0  e.  RR*  /\  k  e.  RR* )  ->  (
0  <_  k  ->  0  <_  k ) )
30 xrltle 11118 . . . . 5  |-  ( ( k  e.  RR*  /\  M  e.  RR* )  ->  (
k  <  M  ->  k  <_  M ) )
3127, 28, 29, 30ixxssixx 11306 . . . 4  |-  ( 0 [,) M )  C_  ( 0 [,] M
)
32 imass2 5199 . . . 4  |-  ( ( 0 [,) M ) 
C_  ( 0 [,] M )  ->  ( `' abs " ( 0 [,) M ) ) 
C_  ( `' abs " ( 0 [,] M
) ) )
3331, 32mp1i 12 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,) M ) ) 
C_  ( `' abs " ( 0 [,] M
) ) )
3425, 33eqsstr3d 3386 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) M )  C_  ( `' abs " ( 0 [,] M ) ) )
35 iccssxr 11370 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
36 pserf.g . . . . . . . 8  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
37 pserf.a . . . . . . . 8  |-  ( ph  ->  A : NN0 --> CC )
38 pserf.r . . . . . . . 8  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
3936, 37, 38radcnvcl 21857 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
4039adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,] +oo ) )
4135, 40sseldi 3349 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
4211simp3d 1002 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
43 xrlelttr 11122 . . . . . 6  |-  ( ( z  e.  RR*  /\  M  e.  RR*  /\  R  e. 
RR* )  ->  (
( z  <_  M  /\  M  <  R )  ->  z  <  R
) )
4427, 28, 43ixxss2 11311 . . . . 5  |-  ( ( R  e.  RR*  /\  M  <  R )  ->  (
0 [,] M ) 
C_  ( 0 [,) R ) )
4541, 42, 44syl2anc 661 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
0 [,] M ) 
C_  ( 0 [,) R ) )
46 imass2 5199 . . . 4  |-  ( ( 0 [,] M ) 
C_  ( 0 [,) R )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  ( `' abs " ( 0 [,) R
) ) )
4745, 46syl 16 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  ( `' abs " ( 0 [,) R
) ) )
4847, 1syl6sseqr 3398 . 2  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  S )
4926, 34, 483jca 1168 1  |-  ( (
ph  /\  a  e.  S )  ->  (
a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) M )  /\  (
0 ( ball `  ( abs  o.  -  ) ) M )  C_  ( `' abs " ( 0 [,] M ) )  /\  ( `' abs " ( 0 [,] M
) )  C_  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2714    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   `'ccnv 4834   dom cdm 4835   "cima 4838    o. ccom 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   supcsup 7682   CCcc 9272   RRcr 9273   0cc0 9274    + caddc 9277    x. cmul 9279   +oocpnf 9407   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587   NN0cn0 10571   RR+crp 10983   [,)cico 11294   [,]cicc 11295    seqcseq 11798   ^cexp 11857   abscabs 12715    ~~> cli 12954   sum_csu 13155   ballcbl 17778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-xadd 11082  df-ico 11298  df-icc 11299  df-fz 11430  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787
This theorem is referenced by:  psercn  21866  pserdvlem2  21868  pserdv  21869
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