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Theorem psercnlem2 20200
Description: Lemma for psercn 20202. (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 5157 . . . . . . . 8  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
3 absf 12061 . . . . . . . . 9  |-  abs : CC
--> RR
43fdmi 5529 . . . . . . . 8  |-  dom  abs  =  CC
52, 4sseqtri 3316 . . . . . . 7  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
61, 5eqsstri 3314 . . . . . 6  |-  S  C_  CC
76a1i 11 . . . . 5  |-  ( ph  ->  S  C_  CC )
87sselda 3284 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
98abscld 12158 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
108absge0d 12166 . . . . 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 970 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
13 0re 9017 . . . . . 6  |-  0  e.  RR
1411simp1d 969 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
1514rpxrd 10574 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR* )
16 elico2 10899 . . . . . 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 645 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  ( 0 [,) M )  <->  ( ( abs `  a )  e.  RR  /\  0  <_ 
( abs `  a
)  /\  ( abs `  a )  <  M
) ) )
189, 10, 12, 17mpbir3and 1137 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  ( 0 [,) M
) )
19 ffn 5524 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
20 elpreima 5782 . . . . 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 646 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( `' abs " (
0 [,) M ) ) )
23 eqid 2380 . . . . 5  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2423cnbl0 18672 . . . 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 2456 . 2  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) M ) )
27 df-ico 10847 . . . . 5  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
28 df-icc 10848 . . . . 5  |-  [,]  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <_  v ) } )
29 idd 22 . . . . 5  |-  ( ( 0  e.  RR*  /\  k  e.  RR* )  ->  (
0  <_  k  ->  0  <_  k ) )
30 xrltle 10667 . . . . 5  |-  ( ( k  e.  RR*  /\  M  e.  RR* )  ->  (
k  <  M  ->  k  <_  M ) )
3127, 28, 29, 30ixxssixx 10855 . . . 4  |-  ( 0 [,) M )  C_  ( 0 [,] M
)
32 imass2 5173 . . . 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 3319 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) M )  C_  ( `' abs " ( 0 [,] M ) ) )
35 iccssxr 10918 . . . . . 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 20193 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
4039adantr 452 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,]  +oo ) )
4135, 40sseldi 3282 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
4211simp3d 971 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
43 xrlelttr 10671 . . . . . 6  |-  ( ( z  e.  RR*  /\  M  e.  RR*  /\  R  e. 
RR* )  ->  (
( z  <_  M  /\  M  <  R )  ->  z  <  R
) )
4427, 28, 43ixxss2 10860 . . . . 5  |-  ( ( R  e.  RR*  /\  M  <  R )  ->  (
0 [,] M ) 
C_  ( 0 [,) R ) )
4541, 42, 44syl2anc 643 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
0 [,] M ) 
C_  ( 0 [,) R ) )
46 imass2 5173 . . . 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 3331 . 2  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  S )
4926, 34, 483jca 1134 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2646    C_ wss 3256   class class class wbr 4146    e. cmpt 4200   `'ccnv 4810   dom cdm 4811   "cima 4814    o. ccom 4815    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   supcsup 7373   CCcc 8914   RRcr 8915   0cc0 8916    + caddc 8919    x. cmul 8921    +oocpnf 9043   RR*cxr 9045    < clt 9046    <_ cle 9047    - cmin 9216   NN0cn0 10146   RR+crp 10537   [,)cico 10843   [,]cicc 10844    seq cseq 11243   ^cexp 11302   abscabs 11959    ~~> cli 12198   sum_csu 12399   ballcbl 16607
This theorem is referenced by:  psercn  20202  pserdvlem2  20204  pserdv  20205
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-xadd 10636  df-ico 10847  df-icc 10848  df-fz 10969  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-xmet 16612  df-met 16613  df-bl 16614
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