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Theorem psercnlem2 19800
Description: Lemma for psercn 19802. (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 5033 . . . . . . . 8  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
3 absf 11821 . . . . . . . . 9  |-  abs : CC
--> RR
43fdmi 5394 . . . . . . . 8  |-  dom  abs  =  CC
52, 4sseqtri 3210 . . . . . . 7  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
61, 5eqsstri 3208 . . . . . 6  |-  S  C_  CC
76a1i 10 . . . . 5  |-  ( ph  ->  S  C_  CC )
87sselda 3180 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
98abscld 11918 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
108absge0d 11926 . . . . 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 968 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
13 0re 8838 . . . . . 6  |-  0  e.  RR
1411simp1d 967 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
1514rpxrd 10391 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR* )
16 elico2 10714 . . . . . 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 644 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  ( 0 [,) M )  <->  ( ( abs `  a )  e.  RR  /\  0  <_ 
( abs `  a
)  /\  ( abs `  a )  <  M
) ) )
189, 10, 12, 17mpbir3and 1135 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  ( 0 [,) M
) )
19 ffn 5389 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
20 elpreima 5645 . . . . 5  |-  ( abs 
Fn  CC  ->  ( a  e.  ( `' abs " ( 0 [,) M
) )  <->  ( a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) M ) ) ) )
213, 19, 20mp2b 9 . . . 4  |-  ( a  e.  ( `' abs " ( 0 [,) M
) )  <->  ( a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) M ) ) )
228, 18, 21sylanbrc 645 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( `' abs " (
0 [,) M ) ) )
23 eqid 2283 . . . . 5  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2423cnbl0 18283 . . . 4  |-  ( M  e.  RR*  ->  ( `' abs " ( 0 [,) M ) )  =  ( 0 (
ball `  ( abs  o. 
-  ) ) M ) )
2515, 24syl 15 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,) M ) )  =  ( 0 (
ball `  ( abs  o. 
-  ) ) M ) )
2622, 25eleqtrd 2359 . 2  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) M ) )
27 df-ico 10662 . . . . 5  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
28 df-icc 10663 . . . . 5  |-  [,]  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <_  v ) } )
29 idd 21 . . . . 5  |-  ( ( 0  e.  RR*  /\  k  e.  RR* )  ->  (
0  <_  k  ->  0  <_  k ) )
30 xrltle 10483 . . . . 5  |-  ( ( k  e.  RR*  /\  M  e.  RR* )  ->  (
k  <  M  ->  k  <_  M ) )
3127, 28, 29, 30ixxssixx 10670 . . . 4  |-  ( 0 [,) M )  C_  ( 0 [,] M
)
32 imass2 5049 . . . 4  |-  ( ( 0 [,) M ) 
C_  ( 0 [,] M )  ->  ( `' abs " ( 0 [,) M ) ) 
C_  ( `' abs " ( 0 [,] M
) ) )
3331, 32mp1i 11 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,) M ) ) 
C_  ( `' abs " ( 0 [,] M
) ) )
3425, 33eqsstr3d 3213 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) M )  C_  ( `' abs " ( 0 [,] M ) ) )
35 iccssxr 10732 . . . . . 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 19793 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
4039adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,]  +oo ) )
4135, 40sseldi 3178 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
4211simp3d 969 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
43 xrlelttr 10487 . . . . . 6  |-  ( ( z  e.  RR*  /\  M  e.  RR*  /\  R  e. 
RR* )  ->  (
( z  <_  M  /\  M  <  R )  ->  z  <  R
) )
4427, 28, 43ixxss2 10675 . . . . 5  |-  ( ( R  e.  RR*  /\  M  <  R )  ->  (
0 [,] M ) 
C_  ( 0 [,) R ) )
4541, 42, 44syl2anc 642 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
0 [,] M ) 
C_  ( 0 [,) R ) )
46 imass2 5049 . . . 4  |-  ( ( 0 [,] M ) 
C_  ( 0 [,) R )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  ( `' abs " ( 0 [,) R
) ) )
4745, 46syl 15 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  ( `' abs " ( 0 [,) R
) ) )
4847, 1syl6sseqr 3225 . 2  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  S )
4926, 34, 483jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   "cima 4692    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   NN0cn0 9965   RR+crp 10354   [,)cico 10658   [,]cicc 10659    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958   sum_csu 12158   ballcbl 16371
This theorem is referenced by:  psercn  19802  pserdvlem2  19804  pserdv  19805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-xadd 10453  df-ico 10662  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-xmet 16373  df-met 16374  df-bl 16375
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