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Theorem psercnlem1 21895
Description: Lemma for psercn 21896. (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 5194 . . . . . . . . . . . 12  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
4 absf 12830 . . . . . . . . . . . . 13  |-  abs : CC
--> RR
54fdmi 5569 . . . . . . . . . . . 12  |-  dom  abs  =  CC
63, 5sseqtri 3393 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
72, 6eqsstri 3391 . . . . . . . . . 10  |-  S  C_  CC
87a1i 11 . . . . . . . . 9  |-  ( ph  ->  S  C_  CC )
98sselda 3361 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
109abscld 12927 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
11 readdcl 9370 . . . . . . 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 10576 . . . . 5  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  (
( ( abs `  a
)  +  R )  /  2 )  e.  RR )
14 peano2re 9547 . . . . . . 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 3826 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  if ( R  e.  RR ,  ( ( ( abs `  a )  +  R )  / 
2 ) ,  ( ( abs `  a
)  +  1 ) )  e.  RR )
181, 17syl5eqel 2527 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
19 0re 9391 . . . . 5  |-  0  e.  RR
2019a1i 11 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
219absge0d 12935 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
22 breq2 4301 . . . . . 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 4301 . . . . . 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 2533 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( `' abs " (
0 [,) R ) ) )
26 ffn 5564 . . . . . . . . . . . . 13  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
27 elpreima 5828 . . . . . . . . . . . . 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 11383 . . . . . . . . . . . 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 21887 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
3635adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  ( 0 [,] +oo ) )
3731, 36sseldi 3359 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
38 elico2 11364 . . . . . . . . . . 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 1002 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
R )
4241adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  S )  /\  R  e.  RR )  ->  ( abs `  a )  < 
R )
43 avglt1 10567 . . . . . . . 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 10268 . . . . . . 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 3829 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) ) )
4948, 1syl6breqr 4337 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
5020, 10, 18, 21, 49lelttrd 9534 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  0  <  M )
5118, 50elrpd 11030 . 2  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
52 breq1 4300 . . . 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 4300 . . . 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 10568 . . . . . 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 9438 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  1 )  e.  RR* )
58 xrlenlt 9447 . . . . . . . 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 9435 . . . . . . . . . . . . 13  |-  0  e.  RR*
61 pnfxr 11097 . . . . . . . . . . . . 13  |- +oo  e.  RR*
62 elicc1 11349 . . . . . . . . . . . . 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 1001 . . . . . . . . . 10  |-  ( ph  ->  0  <_  R )
6665adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  R )
67 ge0gtmnf 11149 . . . . . . . . 9  |-  ( ( R  e.  RR*  /\  0  <_  R )  -> -oo  <  R )
6837, 66, 67syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  a  e.  S )  -> -oo  <  R )
69 xrre 11146 . . . . . . . . 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 1217 . . . . . . 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 3829 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  if ( R  e.  RR ,  ( ( ( abs `  a )  +  R )  / 
2 ) ,  ( ( abs `  a
)  +  1 ) )  <  R )
761, 75syl5eqbr 4330 . 2  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
7751, 49, 763jca 1168 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 965    = wceq 1369    e. wcel 1756   {crab 2724    C_ wss 3333   ifcif 3796   class class class wbr 4297    e. cmpt 4355   `'ccnv 4844   dom cdm 4845   "cima 4848    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   supcsup 7695   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292   +oocpnf 9420   -oocmnf 9421   RR*cxr 9422    < clt 9423    <_ cle 9424    / cdiv 9998   2c2 10376   NN0cn0 10584   RR+crp 10996   [,)cico 11307   [,]cicc 11308    seqcseq 11811   ^cexp 11870   abscabs 12728    ~~> cli 12967   sum_csu 13168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-ico 11311  df-icc 11312  df-fz 11443  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971
This theorem is referenced by:  psercn  21896  pserdvlem1  21897  pserdvlem2  21898  pserdv  21899
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