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Theorem psercnlem2 22945
Description: Lemma for psercn 22947. (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  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 5367 . . . . . . . 8  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
3 absf 13182 . . . . . . . . 9  |-  abs : CC
--> RR
43fdmi 5742 . . . . . . . 8  |-  dom  abs  =  CC
52, 4sseqtri 3531 . . . . . . 7  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
61, 5eqsstri 3529 . . . . . 6  |-  S  C_  CC
76a1i 11 . . . . 5  |-  ( ph  ->  S  C_  CC )
87sselda 3499 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
98abscld 13279 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
108absge0d 13287 . . . . 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 1009 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
13 0re 9613 . . . . . 6  |-  0  e.  RR
1411simp1d 1008 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
1514rpxrd 11282 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR* )
16 elico2 11613 . . . . . 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 1179 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  ( 0 [,) M
) )
19 ffn 5737 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
20 elpreima 6008 . . . . 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 2457 . . . . 5  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2423cnbl0 21407 . . . 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 2547 . 2  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) M ) )
27 icossicc 11636 . . . 4  |-  ( 0 [,) M )  C_  ( 0 [,] M
)
28 imass2 5382 . . . 4  |-  ( ( 0 [,) M ) 
C_  ( 0 [,] M )  ->  ( `' abs " ( 0 [,) M ) ) 
C_  ( `' abs " ( 0 [,] M
) ) )
2927, 28mp1i 12 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,) M ) ) 
C_  ( `' abs " ( 0 [,] M
) ) )
3025, 29eqsstr3d 3534 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) M )  C_  ( `' abs " ( 0 [,] M ) ) )
31 iccssxr 11632 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
32 pserf.g . . . . . . . 8  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
33 pserf.a . . . . . . . 8  |-  ( ph  ->  A : NN0 --> CC )
34 pserf.r . . . . . . . 8  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
3532, 33, 34radcnvcl 22938 . . . . . . 7  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
3635adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,] +oo ) )
3731, 36sseldi 3497 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
3811simp3d 1010 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
39 df-ico 11560 . . . . . 6  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
40 df-icc 11561 . . . . . 6  |-  [,]  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <_  v ) } )
41 xrlelttr 11384 . . . . . 6  |-  ( ( z  e.  RR*  /\  M  e.  RR*  /\  R  e. 
RR* )  ->  (
( z  <_  M  /\  M  <  R )  ->  z  <  R
) )
4239, 40, 41ixxss2 11573 . . . . 5  |-  ( ( R  e.  RR*  /\  M  <  R )  ->  (
0 [,] M ) 
C_  ( 0 [,) R ) )
4337, 38, 42syl2anc 661 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
0 [,] M ) 
C_  ( 0 [,) R ) )
44 imass2 5382 . . . 4  |-  ( ( 0 [,] M ) 
C_  ( 0 [,) R )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  ( `' abs " ( 0 [,) R
) ) )
4543, 44syl 16 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  ( `' abs " ( 0 [,) R
) ) )
4645, 1syl6sseqr 3546 . 2  |-  ( (
ph  /\  a  e.  S )  ->  ( `' abs " ( 0 [,] M ) ) 
C_  S )
4726, 30, 463jca 1176 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 973    = wceq 1395    e. wcel 1819   {crab 2811    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   "cima 5011    o. ccom 5012    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512    x. cmul 9514   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824   NN0cn0 10816   RR+crp 11245   [,)cico 11556   [,]cicc 11557    seqcseq 12110   ^cexp 12169   abscabs 13079    ~~> cli 13319   sum_csu 13520   ballcbl 18532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-xadd 11344  df-ico 11560  df-icc 11561  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541
This theorem is referenced by:  psercn  22947  pserdvlem2  22949  pserdv  22950
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