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Theorem pntrlog2bndlem6a 20731
Description: Lemma for pntrlog2bndlem6 20732. (Contributed by Mario Carneiro, 7-Jun-2016.)
Hypotheses
Ref Expression
pntsval.1  |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i )  x.  ( ( log `  i
)  +  (ψ `  ( a  /  i
) ) ) ) )
pntrlog2bnd.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
pntrlog2bnd.t  |-  T  =  ( a  e.  RR  |->  if ( a  e.  RR+ ,  ( a  x.  ( log `  a ) ) ,  0 ) )
pntrlog2bndlem5.1  |-  ( ph  ->  B  e.  RR+ )
pntrlog2bndlem5.2  |-  ( ph  ->  A. y  e.  RR+  ( abs `  ( ( R `  y )  /  y ) )  <_  B )
pntrlog2bndlem6.1  |-  ( ph  ->  A  e.  RR )
pntrlog2bndlem6.2  |-  ( ph  ->  1  <_  A )
Assertion
Ref Expression
pntrlog2bndlem6a  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
1 ... ( |_ `  x ) )  =  ( ( 1 ... ( |_ `  (
x  /  A ) ) )  u.  (
( ( |_ `  ( x  /  A
) )  +  1 ) ... ( |_
`  x ) ) ) )
Distinct variable groups:    i, a, x, y, A    x, B, y    ph, x    x, S, y    x, R, y
Allowed substitution hints:    ph( y, i, a)    B( i, a)    R( i, a)    S( i, a)    T( x, y, i, a)

Proof of Theorem pntrlog2bndlem6a
StepHypRef Expression
1 elioore 10686 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR )
21adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  RR )
3 1rp 10358 . . . . . . . 8  |-  1  e.  RR+
43a1i 10 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  e.  RR+ )
54rpred 10390 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  e.  RR )
6 eliooord 10710 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
1  <  x  /\  x  <  +oo ) )
76adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
1  <  x  /\  x  <  +oo ) )
87simpld 445 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <  x )
95, 2, 8ltled 8967 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <_  x )
102, 4, 9rpgecld 10425 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  RR+ )
11 pntrlog2bndlem6.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
123a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  RR+ )
13 pntrlog2bndlem6.2 . . . . . . . 8  |-  ( ph  ->  1  <_  A )
1411, 12, 13rpgecld 10425 . . . . . . 7  |-  ( ph  ->  A  e.  RR+ )
1514adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  A  e.  RR+ )
1610, 15rpdivcld 10407 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  e.  RR+ )
1716rprege0d 10397 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( x  /  A
)  e.  RR  /\  0  <_  ( x  /  A ) ) )
18 flge0nn0 10948 . . . 4  |-  ( ( ( x  /  A
)  e.  RR  /\  0  <_  ( x  /  A ) )  -> 
( |_ `  (
x  /  A ) )  e.  NN0 )
19 nn0p1nn 10003 . . . 4  |-  ( ( |_ `  ( x  /  A ) )  e.  NN0  ->  ( ( |_ `  ( x  /  A ) )  +  1 )  e.  NN )
2017, 18, 193syl 18 . . 3  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( |_ `  (
x  /  A ) )  +  1 )  e.  NN )
21 nnuz 10263 . . 3  |-  NN  =  ( ZZ>= `  1 )
2220, 21syl6eleq 2373 . 2  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( |_ `  (
x  /  A ) )  +  1 )  e.  ( ZZ>= `  1
) )
2316rpred 10390 . . 3  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  e.  RR )
2410rpge0d 10394 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  0  <_  x )
2513adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <_  A )
264, 15, 2, 24, 25lediv2ad 10412 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  <_  ( x  / 
1 ) )
272recnd 8861 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  CC )
2827div1d 9528 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  1 )  =  x )
2926, 28breqtrd 4047 . . 3  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  <_  x )
30 flword2 10943 . . 3  |-  ( ( ( x  /  A
)  e.  RR  /\  x  e.  RR  /\  (
x  /  A )  <_  x )  -> 
( |_ `  x
)  e.  ( ZZ>= `  ( |_ `  ( x  /  A ) ) ) )
3123, 2, 29, 30syl3anc 1182 . 2  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  ( |_ `  x )  e.  ( ZZ>= `  ( |_ `  ( x  /  A
) ) ) )
32 fzsplit2 10815 . 2  |-  ( ( ( ( |_ `  ( x  /  A
) )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  x )  e.  ( ZZ>= `  ( |_ `  ( x  /  A
) ) ) )  ->  ( 1 ... ( |_ `  x
) )  =  ( ( 1 ... ( |_ `  ( x  /  A ) ) )  u.  ( ( ( |_ `  ( x  /  A ) )  +  1 ) ... ( |_ `  x
) ) ) )
3322, 31, 32syl2anc 642 1  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
1 ... ( |_ `  x ) )  =  ( ( 1 ... ( |_ `  (
x  /  A ) ) )  u.  (
( ( |_ `  ( x  /  A
) )  +  1 ) ... ( |_
`  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZ>=cuz 10230   RR+crp 10354   (,)cioo 10656   ...cfz 10782   |_cfl 10924   abscabs 11719   sum_csu 12158   logclog 19912  Λcvma 20329  ψcchp 20330
This theorem is referenced by:  pntrlog2bndlem6  20732
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioo 10660  df-fz 10783  df-fl 10925
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