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Theorem pserdvlem1 23319
Description: Lemma for pserdv 23321. (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 5145 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
3 absf 13339 . . . . . . . . . . 11  |-  abs : CC
--> RR
43fdmi 5689 . . . . . . . . . 10  |-  dom  abs  =  CC
52, 4sseqtri 3434 . . . . . . . . 9  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
61, 5eqsstri 3432 . . . . . . . 8  |-  S  C_  CC
76a1i 11 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
87sselda 3402 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
98abscld 13436 . . . . 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 23317 . . . . . . 7  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
1615simp1d 1017 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
1716rpred 11287 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
189, 17readdcld 9616 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR )
19 0red 9590 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
208absge0d 13444 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
219, 16ltaddrpd 11317 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  M ) )
2219, 9, 18, 20, 21lelttrd 9739 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  0  <  ( ( abs `  a
)  +  M ) )
2318, 22elrpd 11284 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR+ )
2423rphalfcld 11299 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR+ )
2515simp2d 1018 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
M )
26 avglt1 10796 . . . 4  |-  ( ( ( abs `  a
)  e.  RR  /\  M  e.  RR )  ->  ( ( abs `  a
)  <  M  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  M )  /  2 ) ) )
279, 17, 26syl2anc 665 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  <  M  <->  ( abs `  a )  <  (
( ( abs `  a
)  +  M )  /  2 ) ) )
2825, 27mpbid 213 . 2  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( ( abs `  a )  +  M
)  /  2 ) )
2918rehalfcld 10805 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR )
3029rexrd 9636 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )
3117rexrd 9636 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR* )
32 iccssxr 11663 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
3310, 12, 13radcnvcl 23309 . . . . 5  |-  ( ph  ->  R  e.  ( 0 [,] +oo ) )
3432, 33sseldi 3400 . . . 4  |-  ( ph  ->  R  e.  RR* )
3534adantr 466 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  R  e.  RR* )
36 avglt2 10797 . . . . 5  |-  ( ( ( abs `  a
)  e.  RR  /\  M  e.  RR )  ->  ( ( abs `  a
)  <  M  <->  ( (
( abs `  a
)  +  M )  /  2 )  < 
M ) )
379, 17, 36syl2anc 665 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  <  M  <->  ( (
( abs `  a
)  +  M )  /  2 )  < 
M ) )
3825, 37mpbid 213 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  < 
M )
3915simp3d 1019 . . 3  |-  ( (
ph  /\  a  e.  S )  ->  M  <  R )
4030, 31, 35, 38, 39xrlttrd 11402 . 2  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  < 
R )
4124, 28, 403jca 1185 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   {crab 2713    C_ wss 3374   ifcif 3849   class class class wbr 4361    |-> cmpt 4420   `'ccnv 4790   dom cdm 4791   "cima 4794   -->wf 5535   ` cfv 5539  (class class class)co 6244   supcsup 7902   CCcc 9483   RRcr 9484   0cc0 9485   1c1 9486    + caddc 9488    x. cmul 9490   +oocpnf 9618   RR*cxr 9620    < clt 9621    / cdiv 10215   2c2 10605   NN0cn0 10815   RR+crp 11248   [,)cico 11583   [,]cicc 11584    seqcseq 12158   ^cexp 12217   abscabs 13236    ~~> cli 13486   sum_csu 13690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-sup 7904  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-ico 11587  df-icc 11588  df-fz 11731  df-seq 12159  df-exp 12218  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-clim 13490
This theorem is referenced by:  pserdvlem2  23320  pserdv  23321
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