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Theorem psercnlem1 19801
Description: Lemma for psercn 19802. (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 5033 . . . . . . . . . . . 12  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
4 absf 11821 . . . . . . . . . . . . 13  |-  abs : CC
--> RR
54fdmi 5394 . . . . . . . . . . . 12  |-  dom  abs  =  CC
63, 5sseqtri 3210 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
72, 6eqsstri 3208 . . . . . . . . . 10  |-  S  C_  CC
87a1i 10 . . . . . . . . 9  |-  ( ph  ->  S  C_  CC )
98sselda 3180 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
109abscld 11918 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
11 readdcl 8820 . . . . . . 7  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  +  R )  e.  RR )
1210, 11sylan 457 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  +  R )  e.  RR )
1312rehalfcld 9958 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( ( abs `  a
)  +  R )  /  2 )  e.  RR )
14 peano2re 8985 . . . . . . 7  |-  ( ( abs `  a )  e.  RR  ->  (
( abs `  a
)  +  1 )  e.  RR )
1510, 14syl 15 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  1 )  e.  RR )
1615adantr 451 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( ( abs `  a
)  +  1 )  e.  RR )
1713, 16ifclda 3592 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  if ( R  e.  RR ,  ( ( ( abs `  a )  +  R )  / 
2 ) ,  ( ( abs `  a
)  +  1 ) )  e.  RR )
181, 17syl5eqel 2367 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
19 0re 8838 . . . . 5  |-  0  e.  RR
2019a1i 10 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
219absge0d 11926 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
22 breq2 4027 . . . . . 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 4027 . . . . . 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 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  S )
2524, 2syl6eleq 2373 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( `' abs " (
0 [,) R ) ) )
26 ffn 5389 . . . . . . . . . . . . 13  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
27 elpreima 5645 . . . . . . . . . . . . 13  |-  ( abs 
Fn  CC  ->  ( a  e.  ( `' abs " ( 0 [,) R
) )  <->  ( a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) R ) ) ) )
284, 26, 27mp2b 9 . . . . . . . . . . . 12  |-  ( a  e.  ( `' abs " ( 0 [,) R
) )  <->  ( a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) R ) ) )
2925, 28sylib 188 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  (
a  e.  CC  /\  ( abs `  a )  e.  ( 0 [,) R ) ) )
3029simprd 449 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  ( 0 [,) R
) )
31 iccssxr 10732 . . . . . . . . . . . 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 19793 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
3635adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,]  +oo ) )
3731, 36sseldi 3178 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
38 elico2 10714 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  ( 0 [,) R )  <->  ( ( abs `  a )  e.  RR  /\  0  <_ 
( abs `  a
)  /\  ( abs `  a )  <  R
) ) )
4030, 39mpbid 201 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  e.  RR  /\  0  <_  ( abs `  a
)  /\  ( abs `  a )  <  R
) )
4140simp3d 969 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
R )
4241adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  ( abs `  a )  < 
R )
43 avglt1 9949 . . . . . . . 8  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  <  R  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  R )  /  2 ) ) )
4410, 43sylan 457 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  <  R  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  R )  /  2 ) ) )
4542, 44mpbid 201 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  ( abs `  a )  < 
( ( ( abs `  a )  +  R
)  /  2 ) )
4610ltp1d 9687 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  1 ) )
4746adantr 451 . . . . . 6  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( abs `  a
)  <  ( ( abs `  a )  +  1 ) )
4822, 23, 45, 47ifbothda 3595 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) ) )
4948, 1syl6breqr 4063 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
5020, 10, 18, 21, 49lelttrd 8974 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  0  <  M )
5118, 50elrpd 10388 . 2  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
52 breq1 4026 . . . 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 4026 . . . 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 9950 . . . . . 6  |-  ( ( ( abs `  a
)  e.  RR  /\  R  e.  RR )  ->  ( ( abs `  a
)  <  R  <->  ( (
( abs `  a
)  +  R )  /  2 )  < 
R ) )
5510, 54sylan 457 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( abs `  a
)  <  R  <->  ( (
( abs `  a
)  +  R )  /  2 )  < 
R ) )
5642, 55mpbid 201 . . . 4  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( ( abs `  a
)  +  R )  /  2 )  < 
R )
5715rexrd 8881 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  1 )  e.  RR* )
58 xrlenlt 8890 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  (
( abs `  a
)  +  1 )  e.  RR* )  ->  ( R  <_  ( ( abs `  a )  +  1 )  <->  -.  ( ( abs `  a )  +  1 )  <  R
) )
5937, 57, 58syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( R  <_  ( ( abs `  a )  +  1 )  <->  -.  ( ( abs `  a )  +  1 )  <  R
) )
60 0xr 8878 . . . . . . . . . . . . 13  |-  0  e.  RR*
61 pnfxr 10455 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
62 elicc1 10700 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR* )  ->  ( R  e.  ( 0 [,]  +oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo )
) )
6360, 61, 62mp2an 653 . . . . . . . . . . . 12  |-  ( R  e.  ( 0 [,] 
+oo )  <->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo )
)
6435, 63sylib 188 . . . . . . . . . . 11  |-  ( ph  ->  ( R  e.  RR*  /\  0  <_  R  /\  R  <_  +oo ) )
6564simp2d 968 . . . . . . . . . 10  |-  ( ph  ->  0  <_  R )
6665adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  R )
67 ge0gtmnf 10501 . . . . . . . . 9  |-  ( ( R  e.  RR*  /\  0  <_  R )  ->  -oo  <  R )
6837, 66, 67syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  -oo  <  R )
69 xrre 10498 . . . . . . . . 9  |-  ( ( ( R  e.  RR*  /\  ( ( abs `  a
)  +  1 )  e.  RR )  /\  (  -oo  <  R  /\  R  <_  ( ( abs `  a )  +  1 ) ) )  ->  R  e.  RR )
7069expr 598 . . . . . . . 8  |-  ( ( ( R  e.  RR*  /\  ( ( abs `  a
)  +  1 )  e.  RR )  /\  -oo 
<  R )  ->  ( R  <_  ( ( abs `  a )  +  1 )  ->  R  e.  RR ) )
7137, 15, 68, 70syl21anc 1181 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( R  <_  ( ( abs `  a )  +  1 )  ->  R  e.  RR ) )
7259, 71sylbird 226 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  ( -.  ( ( abs `  a
)  +  1 )  <  R  ->  R  e.  RR ) )
7372con1d 116 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( -.  R  e.  RR  ->  ( ( abs `  a
)  +  1 )  <  R ) )
7473imp 418 . . . 4  |-  ( ( ( ph  /\  a  e.  S )  /\  -.  R  e.  RR )  ->  ( ( abs `  a
)  +  1 )  <  R )
7552, 53, 56, 74ifbothda 3595 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  if ( R  e.  RR ,  ( ( ( abs `  a )  +  R )  / 
2 ) ,  ( ( abs `  a
)  +  1 ) )  <  R )
761, 75syl5eqbr 4056 . 2  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
7751, 49, 763jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868    / cdiv 9423   2c2 9795   NN0cn0 9965   RR+crp 10354   [,)cico 10658   [,]cicc 10659    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  psercn  19802  pserdvlem1  19803  pserdvlem2  19804  pserdv  19805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
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