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Theorem psercnlem1 20342
Description: Lemma for psercn 20343. (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 ) )
psercn.m  |-  M  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )
Assertion
Ref Expression
psercnlem1  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  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 psercnlem1
StepHypRef Expression
1 psercn.m . . . 4  |-  M  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )
2 psercn.s . . . . . . . . . . 11  |-  S  =  ( `' abs " (
0 [,) R ) )
3 cnvimass 5225 . . . . . . . . . . . 12  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
4 absf 12142 . . . . . . . . . . . . 13  |-  abs : CC
--> RR
54fdmi 5597 . . . . . . . . . . . 12  |-  dom  abs  =  CC
63, 5sseqtri 3381 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
72, 6eqsstri 3379 . . . . . . . . . 10  |-  S  C_  CC
87a1i 11 . . . . . . . . 9  |-  ( ph  ->  S  C_  CC )
98sselda 3349 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
109abscld 12239 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
11 readdcl 9074 . . . . . . 7  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  +  R )  e.  RR )
1210, 11sylan 459 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  +  R )  e.  RR )
1312rehalfcld 10215 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( ( abs `  a
)  +  R )  /  2 )  e.  RR )
14 peano2re 9240 . . . . . . 7  |-  ( ( abs `  a )  e.  RR  ->  (
( abs `  a
)  +  1 )  e.  RR )
1510, 14syl 16 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  1 )  e.  RR )
1615adantr 453 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( ( abs `  a
)  +  1 )  e.  RR )
1713, 16ifclda 3767 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  if ( R  e.  RR ,  ( ( ( abs `  a )  +  R )  / 
2 ) ,  ( ( abs `  a
)  +  1 ) )  e.  RR )
181, 17syl5eqel 2521 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
19 0re 9092 . . . . 5  |-  0  e.  RR
2019a1i 11 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
219absge0d 12247 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
22 breq2 4217 . . . . . 6  |-  ( ( ( ( abs `  a
)  +  R )  /  2 )  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )  ->  ( ( abs `  a )  < 
( ( ( abs `  a )  +  R
)  /  2 )  <-> 
( abs `  a
)  <  if ( R  e.  RR , 
( ( ( abs `  a )  +  R
)  /  2 ) ,  ( ( abs `  a )  +  1 ) ) ) )
23 breq2 4217 . . . . . 6  |-  ( ( ( abs `  a
)  +  1 )  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )  ->  ( ( abs `  a )  < 
( ( abs `  a
)  +  1 )  <-> 
( abs `  a
)  <  if ( R  e.  RR , 
( ( ( abs `  a )  +  R
)  /  2 ) ,  ( ( abs `  a )  +  1 ) ) ) )
24 simpr 449 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  S )
2524, 2syl6eleq 2527 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( `' abs " (
0 [,) R ) ) )
26 ffn 5592 . . . . . . . . . . . . 13  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
27 elpreima 5851 . . . . . . . . . . . . 13  |-  ( abs 
Fn  CC  ->  ( a  e.  ( `' abs " ( 0 [,) R
) )  <->  ( a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) R ) ) ) )
284, 26, 27mp2b 10 . . . . . . . . . . . 12  |-  ( a  e.  ( `' abs " ( 0 [,) R
) )  <->  ( a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) R ) ) )
2925, 28sylib 190 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  (
a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) R ) ) )
3029simprd 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  ( 0 [,) R
) )
31 iccssxr 10994 . . . . . . . . . . . 12  |-  ( 0 [,]  +oo )  C_  RR*
32 pserf.g . . . . . . . . . . . . . 14  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
33 pserf.a . . . . . . . . . . . . . 14  |-  ( ph  ->  A : NN0 --> CC )
34 pserf.r . . . . . . . . . . . . . 14  |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
3532, 33, 34radcnvcl 20334 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
3635adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,]  +oo ) )
3731, 36sseldi 3347 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
38 elico2 10975 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  R  e.  RR* )  -> 
( ( abs `  a
)  e.  ( 0 [,) R )  <->  ( ( abs `  a )  e.  RR  /\  0  <_ 
( abs `  a
)  /\  ( abs `  a )  <  R
) ) )
3919, 37, 38sylancr 646 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  ( 0 [,) R )  <->  ( ( abs `  a )  e.  RR  /\  0  <_ 
( abs `  a
)  /\  ( abs `  a )  <  R
) ) )
4030, 39mpbid 203 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  RR  /\  0  <_  ( abs `  a
)  /\  ( abs `  a )  <  R
) )
4140simp3d 972 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
R )
4241adantr 453 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  ( abs `  a )  < 
R )
43 avglt1 10206 . . . . . . . 8  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  <  R  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  R )  /  2 ) ) )
4410, 43sylan 459 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  <  R  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  R )  /  2 ) ) )
4542, 44mpbid 203 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  ( abs `  a )  < 
( ( ( abs `  a )  +  R
)  /  2 ) )
4610ltp1d 9942 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  1 ) )
4746adantr 453 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( abs `  a
)  <  ( ( abs `  a )  +  1 ) )
4822, 23, 45, 47ifbothda 3770 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) ) )
4948, 1syl6breqr 4253 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
5020, 10, 18, 21, 49lelttrd 9229 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  0  <  M )
5118, 50elrpd 10647 . 2  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
52 breq1 4216 . . . 4  |-  ( ( ( ( abs `  a
)  +  R )  /  2 )  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )  ->  ( (
( ( abs `  a
)  +  R )  /  2 )  < 
R  <->  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )  <  R ) )
53 breq1 4216 . . . 4  |-  ( ( ( abs `  a
)  +  1 )  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )  ->  ( (
( abs `  a
)  +  1 )  <  R  <->  if ( R  e.  RR , 
( ( ( abs `  a )  +  R
)  /  2 ) ,  ( ( abs `  a )  +  1 ) )  <  R
) )
54 avglt2 10207 . . . . . 6  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  <  R  <->  ( (
( abs `  a
)  +  R )  /  2 )  < 
R ) )
5510, 54sylan 459 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  <  R  <->  ( (
( abs `  a
)  +  R )  /  2 )  < 
R ) )
5642, 55mpbid 203 . . . 4  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( ( abs `  a
)  +  R )  /  2 )  < 
R )
5715rexrd 9135 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  1 )  e.  RR* )
58 xrlenlt 9144 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  (
( abs `  a
)  +  1 )  e.  RR* )  ->  ( R  <_  ( ( abs `  a )  +  1 )  <->  -.  ( ( abs `  a )  +  1 )  <  R
) )
5937, 57, 58syl2anc 644 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( R  <_  ( ( abs `  a )  +  1 )  <->  -.  ( ( abs `  a )  +  1 )  <  R
) )
60 0xr 9132 . . . . . . . . . . . . 13  |-  0  e.  RR*
61 pnfxr 10714 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
62 elicc1 10961 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR* )  ->  ( R  e.  ( 0 [,]  +oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo )
) )
6360, 61, 62mp2an 655 . . . . . . . . . . . 12  |-  ( R  e.  ( 0 [,] 
+oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo )
)
6435, 63sylib 190 . . . . . . . . . . 11  |-  ( ph  ->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo ) )
6564simp2d 971 . . . . . . . . . 10  |-  ( ph  ->  0  <_  R )
6665adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  R )
67 ge0gtmnf 10761 . . . . . . . . 9  |-  ( ( R  e.  RR*  /\  0  <_  R )  ->  -oo  <  R )
6837, 66, 67syl2anc 644 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  -oo  <  R )
69 xrre 10758 . . . . . . . . 9  |-  ( ( ( R  e.  RR*  /\  ( ( abs `  a
)  +  1 )  e.  RR )  /\  (  -oo  <  R  /\  R  <_  ( ( abs `  a )  +  1 ) ) )  ->  R  e.  RR )
7069expr 600 . . . . . . . 8  |-  ( ( ( R  e.  RR*  /\  ( ( abs `  a
)  +  1 )  e.  RR )  /\  -oo 
<  R )  ->  ( R  <_  ( ( abs `  a )  +  1 )  ->  R  e.  RR ) )
7137, 15, 68, 70syl21anc 1184 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( R  <_  ( ( abs `  a )  +  1 )  ->  R  e.  RR ) )
7259, 71sylbird 228 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  ( -.  ( ( abs `  a
)  +  1 )  <  R  ->  R  e.  RR ) )
7372con1d 119 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( -.  R  e.  RR  ->  ( ( abs `  a
)  +  1 )  <  R ) )
7473imp 420 . . . 4  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( ( abs `  a
)  +  1 )  <  R )
7552, 53, 56, 74ifbothda 3770 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  if ( R  e.  RR ,  ( ( ( abs `  a )  +  R )  / 
2 ) ,  ( ( abs `  a
)  +  1 ) )  <  R )
761, 75syl5eqbr 4246 . 2  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
7751, 49, 763jca 1135 1  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2710    C_ wss 3321   ifcif 3740   class class class wbr 4213    e. cmpt 4267   `'ccnv 4878   dom cdm 4879   "cima 4882    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082   supcsup 7446   CCcc 8989   RRcr 8990   0cc0 8991   1c1 8992    + caddc 8994    x. cmul 8996    +oocpnf 9118    -oocmnf 9119   RR*cxr 9120    < clt 9121    <_ cle 9122    / cdiv 9678   2c2 10050   NN0cn0 10222   RR+crp 10613   [,)cico 10919   [,]cicc 10920    seq cseq 11324   ^cexp 11383   abscabs 12040    ~~> cli 12279   sum_csu 12480
This theorem is referenced by:  psercn  20343  pserdvlem1  20344  pserdvlem2  20345  pserdv  20346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-ico 10923  df-icc 10924  df-fz 11045  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283
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