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Theorem pserdvlem1 22556
Description: Lemma for pserdv 22558. (Contributed by Mario Carneiro, 7-May-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 ) )
psercn.m  |-  M  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )
Assertion
Ref Expression
pserdvlem1  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( ( abs `  a )  +  M
)  /  2 )  e.  RR+  /\  ( abs `  a )  < 
( ( ( abs `  a )  +  M
)  /  2 )  /\  ( ( ( abs `  a )  +  M )  / 
2 )  <  R
) )
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 pserdvlem1
StepHypRef Expression
1 psercn.s . . . . . . . . 9  |-  S  =  ( `' abs " (
0 [,) R ) )
2 cnvimass 5355 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
3 absf 13129 . . . . . . . . . . 11  |-  abs : CC
--> RR
43fdmi 5734 . . . . . . . . . 10  |-  dom  abs  =  CC
52, 4sseqtri 3536 . . . . . . . . 9  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
61, 5eqsstri 3534 . . . . . . . 8  |-  S  C_  CC
76a1i 11 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
87sselda 3504 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
98abscld 13226 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
10 pserf.g . . . . . . . 8  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
11 pserf.f . . . . . . . 8  |-  F  =  ( y  e.  S  |-> 
sum_ j  e.  NN0  ( ( G `  y ) `  j
) )
12 pserf.a . . . . . . . 8  |-  ( ph  ->  A : NN0 --> CC )
13 pserf.r . . . . . . . 8  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
14 psercn.m . . . . . . . 8  |-  M  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )
1510, 11, 12, 13, 1, 14psercnlem1 22554 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
1615simp1d 1008 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
1716rpred 11252 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
189, 17readdcld 9619 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR )
19 0red 9593 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
208absge0d 13234 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
219, 16ltaddrpd 11281 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  M ) )
2219, 9, 18, 20, 21lelttrd 9735 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  <  ( ( abs `  a
)  +  M ) )
2318, 22elrpd 11250 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR+ )
2423rphalfcld 11264 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR+ )
2515simp2d 1009 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
26 avglt1 10772 . . . 4  |-  ( ( ( abs `  a
)  e.  RR  /\  M  e.  RR )  ->  ( ( abs `  a
)  <  M  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  M )  /  2 ) ) )
279, 17, 26syl2anc 661 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  <  M  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  M )  /  2 ) ) )
2825, 27mpbid 210 . 2  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( ( abs `  a )  +  M
)  /  2 ) )
2918rehalfcld 10781 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR )
3029rexrd 9639 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )
3117rexrd 9639 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR* )
32 iccssxr 11603 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
3310, 12, 13radcnvcl 22546 . . . . 5  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
3432, 33sseldi 3502 . . . 4  |-  ( ph  ->  R  e.  RR* )
3534adantr 465 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
36 avglt2 10773 . . . . 5  |-  ( ( ( abs `  a
)  e.  RR  /\  M  e.  RR )  ->  ( ( abs `  a
)  <  M  <->  ( (
( abs `  a
)  +  M )  /  2 )  < 
M ) )
379, 17, 36syl2anc 661 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  <  M  <->  ( (
( abs `  a
)  +  M )  /  2 )  < 
M ) )
3825, 37mpbid 210 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  < 
M )
3915simp3d 1010 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
4030, 31, 35, 38, 39xrlttrd 11358 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  < 
R )
4124, 28, 403jca 1176 1  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( ( abs `  a )  +  M
)  /  2 )  e.  RR+  /\  ( abs `  a )  < 
( ( ( abs `  a )  +  M
)  /  2 )  /\  ( ( ( abs `  a )  +  M )  / 
2 )  <  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   "cima 5002   -->wf 5582   ` cfv 5586  (class class class)co 6282   supcsup 7896   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493   +oocpnf 9621   RR*cxr 9623    < clt 9624    / cdiv 10202   2c2 10581   NN0cn0 10791   RR+crp 11216   [,)cico 11527   [,]cicc 11528    seqcseq 12071   ^cexp 12130   abscabs 13026    ~~> cli 13266   sum_csu 13467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-ico 11531  df-icc 11532  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270
This theorem is referenced by:  pserdvlem2  22557  pserdv  22558
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