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Theorem pserdvlem1 22018
Description: Lemma for pserdv 22020. (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 5290 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
3 absf 12936 . . . . . . . . . . 11  |-  abs : CC
--> RR
43fdmi 5665 . . . . . . . . . 10  |-  dom  abs  =  CC
52, 4sseqtri 3489 . . . . . . . . 9  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
61, 5eqsstri 3487 . . . . . . . 8  |-  S  C_  CC
76a1i 11 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
87sselda 3457 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
98abscld 13033 . . . . 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 22016 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
1615simp1d 1000 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
1716rpred 11131 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
189, 17readdcld 9517 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR )
19 0red 9491 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
208absge0d 13041 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
219, 16ltaddrpd 11160 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  M ) )
2219, 9, 18, 20, 21lelttrd 9633 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  <  ( ( abs `  a
)  +  M ) )
2318, 22elrpd 11129 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR+ )
2423rphalfcld 11143 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR+ )
2515simp2d 1001 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
26 avglt1 10666 . . . 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 10675 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR )
3029rexrd 9537 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )
3117rexrd 9537 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR* )
32 iccssxr 11482 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
3310, 12, 13radcnvcl 22008 . . . . 5  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
3432, 33sseldi 3455 . . . 4  |-  ( ph  ->  R  e.  RR* )
3534adantr 465 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
36 avglt2 10667 . . . . 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 1002 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
4030, 31, 35, 38, 39xrlttrd 11237 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  < 
R )
4124, 28, 403jca 1168 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 965    = wceq 1370    e. wcel 1758   {crab 2799    C_ wss 3429   ifcif 3892   class class class wbr 4393    |-> cmpt 4451   `'ccnv 4940   dom cdm 4941   "cima 4944   -->wf 5515   ` cfv 5519  (class class class)co 6193   supcsup 7794   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391   +oocpnf 9519   RR*cxr 9521    < clt 9522    / cdiv 10097   2c2 10475   NN0cn0 10683   RR+crp 11095   [,)cico 11406   [,]cicc 11407    seqcseq 11916   ^cexp 11975   abscabs 12834    ~~> cli 13073   sum_csu 13274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-ico 11410  df-icc 11411  df-fz 11548  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077
This theorem is referenced by:  pserdvlem2  22019  pserdv  22020
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