MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psercnlem1 Structured version   Unicode version

Theorem psercnlem1 22945
Description: Lemma for psercn 22946. (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 5367 . . . . . . . . . . . 12  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
4 absf 13181 . . . . . . . . . . . . 13  |-  abs : CC
--> RR
54fdmi 5742 . . . . . . . . . . . 12  |-  dom  abs  =  CC
63, 5sseqtri 3531 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
72, 6eqsstri 3529 . . . . . . . . . 10  |-  S  C_  CC
87a1i 11 . . . . . . . . 9  |-  ( ph  ->  S  C_  CC )
98sselda 3499 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
109abscld 13278 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
11 readdcl 9592 . . . . . . 7  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  +  R )  e.  RR )
1210, 11sylan 471 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  +  R )  e.  RR )
1312rehalfcld 10806 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( ( abs `  a
)  +  R )  /  2 )  e.  RR )
14 peano2re 9770 . . . . . . 7  |-  ( ( abs `  a )  e.  RR  ->  (
( abs `  a
)  +  1 )  e.  RR )
1510, 14syl 16 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  1 )  e.  RR )
1615adantr 465 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( ( abs `  a
)  +  1 )  e.  RR )
1713, 16ifclda 3976 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  if ( R  e.  RR ,  ( ( ( abs `  a )  +  R )  / 
2 ) ,  ( ( abs `  a
)  +  1 ) )  e.  RR )
181, 17syl5eqel 2549 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
19 0re 9613 . . . . 5  |-  0  e.  RR
2019a1i 11 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
219absge0d 13286 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
22 breq2 4460 . . . . . 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 4460 . . . . . 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 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  S )
2524, 2syl6eleq 2555 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( `' abs " (
0 [,) R ) ) )
26 ffn 5737 . . . . . . . . . . . . 13  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
27 elpreima 6008 . . . . . . . . . . . . 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 196 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  (
a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) R ) ) )
3029simprd 463 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  ( 0 [,) R
) )
31 iccssxr 11632 . . . . . . . . . . . 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 22937 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
3635adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,] +oo ) )
3731, 36sseldi 3497 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
38 elico2 11613 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  ( 0 [,) R )  <->  ( ( abs `  a )  e.  RR  /\  0  <_ 
( abs `  a
)  /\  ( abs `  a )  <  R
) ) )
4030, 39mpbid 210 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  RR  /\  0  <_  ( abs `  a
)  /\  ( abs `  a )  <  R
) )
4140simp3d 1010 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
R )
4241adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  ( abs `  a )  < 
R )
43 avglt1 10797 . . . . . . . 8  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  <  R  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  R )  /  2 ) ) )
4410, 43sylan 471 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  <  R  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  R )  /  2 ) ) )
4542, 44mpbid 210 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  ( abs `  a )  < 
( ( ( abs `  a )  +  R
)  /  2 ) )
4610ltp1d 10496 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  1 ) )
4746adantr 465 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( abs `  a
)  <  ( ( abs `  a )  +  1 ) )
4822, 23, 45, 47ifbothda 3979 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) ) )
4948, 1syl6breqr 4496 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
5020, 10, 18, 21, 49lelttrd 9757 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  0  <  M )
5118, 50elrpd 11279 . 2  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
52 breq1 4459 . . . 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 4459 . . . 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 10798 . . . . . 6  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  <  R  <->  ( (
( abs `  a
)  +  R )  /  2 )  < 
R ) )
5510, 54sylan 471 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  <  R  <->  ( (
( abs `  a
)  +  R )  /  2 )  < 
R ) )
5642, 55mpbid 210 . . . 4  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( ( abs `  a
)  +  R )  /  2 )  < 
R )
5715rexrd 9660 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  1 )  e.  RR* )
58 xrlenlt 9669 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  (
( abs `  a
)  +  1 )  e.  RR* )  ->  ( R  <_  ( ( abs `  a )  +  1 )  <->  -.  ( ( abs `  a )  +  1 )  <  R
) )
5937, 57, 58syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( R  <_  ( ( abs `  a )  +  1 )  <->  -.  ( ( abs `  a )  +  1 )  <  R
) )
60 0xr 9657 . . . . . . . . . . . . 13  |-  0  e.  RR*
61 pnfxr 11346 . . . . . . . . . . . . 13  |- +oo  e.  RR*
62 elicc1 11598 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  ( R  e.  ( 0 [,] +oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_ +oo )
) )
6360, 61, 62mp2an 672 . . . . . . . . . . . 12  |-  ( R  e.  ( 0 [,] +oo )  <->  ( R  e. 
RR*  /\  0  <_  R  /\  R  <_ +oo )
)
6435, 63sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_ +oo ) )
6564simp2d 1009 . . . . . . . . . 10  |-  ( ph  ->  0  <_  R )
6665adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  R )
67 ge0gtmnf 11398 . . . . . . . . 9  |-  ( ( R  e.  RR*  /\  0  <_  R )  -> -oo  <  R )
6837, 66, 67syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  -> -oo  <  R )
69 xrre 11395 . . . . . . . . 9  |-  ( ( ( R  e.  RR*  /\  ( ( abs `  a
)  +  1 )  e.  RR )  /\  ( -oo  <  R  /\  R  <_  ( ( abs `  a )  +  1 ) ) )  ->  R  e.  RR )
7069expr 615 . . . . . . . 8  |-  ( ( ( R  e.  RR*  /\  ( ( abs `  a
)  +  1 )  e.  RR )  /\ -oo 
<  R )  ->  ( R  <_  ( ( abs `  a )  +  1 )  ->  R  e.  RR ) )
7137, 15, 68, 70syl21anc 1227 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( R  <_  ( ( abs `  a )  +  1 )  ->  R  e.  RR ) )
7259, 71sylbird 235 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  ( -.  ( ( abs `  a
)  +  1 )  <  R  ->  R  e.  RR ) )
7372con1d 124 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( -.  R  e.  RR  ->  ( ( abs `  a
)  +  1 )  <  R ) )
7473imp 429 . . . 4  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( ( abs `  a
)  +  1 )  <  R )
7552, 53, 56, 74ifbothda 3979 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  if ( R  e.  RR ,  ( ( ( abs `  a )  +  R )  / 
2 ) ,  ( ( abs `  a
)  +  1 ) )  <  R )
761, 75syl5eqbr 4489 . 2  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
7751, 49, 763jca 1176 1  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {crab 2811    C_ wss 3471   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646    / cdiv 10227   2c2 10606   NN0cn0 10816   RR+crp 11245   [,)cico 11556   [,]cicc 11557    seqcseq 12109   ^cexp 12168   abscabs 13078    ~~> cli 13318   sum_csu 13519
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-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-ico 11560  df-icc 11561  df-fz 11698  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322
This theorem is referenced by:  psercn  22946  pserdvlem1  22947  pserdvlem2  22948  pserdv  22949
  Copyright terms: Public domain W3C validator