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Theorem pntrlog2bndlem6a 21276
Description: Lemma for pntrlog2bndlem6 21277. (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 10946 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR )
21adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  RR )
3 1rp 10616 . . . . . . . 8  |-  1  e.  RR+
43a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  e.  RR+ )
54rpred 10648 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  e.  RR )
6 eliooord 10970 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
1  <  x  /\  x  <  +oo ) )
76adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
1  <  x  /\  x  <  +oo ) )
87simpld 446 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <  x )
95, 2, 8ltled 9221 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <_  x )
102, 4, 9rpgecld 10683 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  RR+ )
11 pntrlog2bndlem6.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
123a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  RR+ )
13 pntrlog2bndlem6.2 . . . . . . . 8  |-  ( ph  ->  1  <_  A )
1411, 12, 13rpgecld 10683 . . . . . . 7  |-  ( ph  ->  A  e.  RR+ )
1514adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  A  e.  RR+ )
1610, 15rpdivcld 10665 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  e.  RR+ )
1716rprege0d 10655 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( x  /  A
)  e.  RR  /\  0  <_  ( x  /  A ) ) )
18 flge0nn0 11225 . . . 4  |-  ( ( ( x  /  A
)  e.  RR  /\  0  <_  ( x  /  A ) )  -> 
( |_ `  (
x  /  A ) )  e.  NN0 )
19 nn0p1nn 10259 . . . 4  |-  ( ( |_ `  ( x  /  A ) )  e.  NN0  ->  ( ( |_ `  ( x  /  A ) )  +  1 )  e.  NN )
2017, 18, 193syl 19 . . 3  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( |_ `  (
x  /  A ) )  +  1 )  e.  NN )
21 nnuz 10521 . . 3  |-  NN  =  ( ZZ>= `  1 )
2220, 21syl6eleq 2526 . 2  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( |_ `  (
x  /  A ) )  +  1 )  e.  ( ZZ>= `  1
) )
2316rpred 10648 . . 3  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  e.  RR )
2410rpge0d 10652 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  0  <_  x )
2513adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <_  A )
264, 15, 2, 24, 25lediv2ad 10670 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  <_  ( x  / 
1 ) )
272recnd 9114 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  CC )
2827div1d 9782 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  1 )  =  x )
2926, 28breqtrd 4236 . . 3  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  <_  x )
30 flword2 11220 . . 3  |-  ( ( ( x  /  A
)  e.  RR  /\  x  e.  RR  /\  (
x  /  A )  <_  x )  -> 
( |_ `  x
)  e.  ( ZZ>= `  ( |_ `  ( x  /  A ) ) ) )
3123, 2, 29, 30syl3anc 1184 . 2  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  ( |_ `  x )  e.  ( ZZ>= `  ( |_ `  ( x  /  A
) ) ) )
32 fzsplit2 11076 . 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 643 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 359    = wceq 1652    e. wcel 1725   A.wral 2705    u. cun 3318   ifcif 3739   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    +oocpnf 9117    < clt 9120    <_ cle 9121    - cmin 9291    / cdiv 9677   NNcn 10000   NN0cn0 10221   ZZ>=cuz 10488   RR+crp 10612   (,)cioo 10916   ...cfz 11043   |_cfl 11201   abscabs 12039   sum_csu 12479   logclog 20452  Λcvma 20874  ψcchp 20875
This theorem is referenced by:  pntrlog2bndlem6  21277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ioo 10920  df-fz 11044  df-fl 11202
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