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Theorem psercnlem2 22546
Description: Lemma for psercn 22548. (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 5348 . . . . . . . 8  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
3 absf 13119 . . . . . . . . 9  |-  abs : CC
--> RR
43fdmi 5727 . . . . . . . 8  |-  dom  abs  =  CC
52, 4sseqtri 3529 . . . . . . 7  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
61, 5eqsstri 3527 . . . . . 6  |-  S  C_  CC
76a1i 11 . . . . 5  |-  ( ph  ->  S  C_  CC )
87sselda 3497 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
98abscld 13216 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
108absge0d 13224 . . . . 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 1004 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
13 0re 9585 . . . . . 6  |-  0  e.  RR
1411simp1d 1003 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
1514rpxrd 11246 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR* )
16 elico2 11577 . . . . . 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 1174 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  ( 0 [,) M
) )
19 ffn 5722 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
20 elpreima 5992 . . . . 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 2460 . . . . 5  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2423cnbl0 21009 . . . 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 2550 . 2  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) M ) )
27 df-ico 11524 . . . . 5  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
28 df-icc 11525 . . . . 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 11344 . . . . 5  |-  ( ( k  e.  RR*  /\  M  e.  RR* )  ->  (
k  <  M  ->  k  <_  M ) )
3127, 28, 29, 30ixxssixx 11532 . . . 4  |-  ( 0 [,) M )  C_  ( 0 [,] M
)
32 imass2 5363 . . . 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 3532 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) M )  C_  ( `' abs " ( 0 [,] M ) ) )
35 iccssxr 11596 . . . . . 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 22539 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
4039adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,] +oo ) )
4135, 40sseldi 3495 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
4211simp3d 1005 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
43 xrlelttr 11348 . . . . . 6  |-  ( ( z  e.  RR*  /\  M  e.  RR*  /\  R  e. 
RR* )  ->  (
( z  <_  M  /\  M  <  R )  ->  z  <  R
) )
4427, 28, 43ixxss2 11537 . . . . 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 5363 . . . 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 3544 . 2  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  S )
4926, 34, 483jca 1171 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 968    = wceq 1374    e. wcel 1762   {crab 2811    C_ wss 3469   class class class wbr 4440    |-> cmpt 4498   `'ccnv 4991   dom cdm 4992   "cima 4995    o. ccom 4996    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   supcsup 7889   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    x. cmul 9486   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9794   NN0cn0 10784   RR+crp 11209   [,)cico 11520   [,]cicc 11521    seqcseq 12063   ^cexp 12122   abscabs 13017    ~~> cli 13256   sum_csu 13457   ballcbl 18169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-xadd 11308  df-ico 11524  df-icc 11525  df-fz 11662  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178
This theorem is referenced by:  psercn  22548  pserdvlem2  22550  pserdv  22551
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