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Theorem pntrlog2bndlem6a 20784
Description: Lemma for pntrlog2bndlem6 20785. (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 10733 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR )
21adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  RR )
3 1rp 10405 . . . . . . . 8  |-  1  e.  RR+
43a1i 10 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  e.  RR+ )
54rpred 10437 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  e.  RR )
6 eliooord 10757 . . . . . . . . . 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 9012 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <_  x )
102, 4, 9rpgecld 10472 . . . . . 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 10472 . . . . . . 7  |-  ( ph  ->  A  e.  RR+ )
1514adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  A  e.  RR+ )
1610, 15rpdivcld 10454 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  e.  RR+ )
1716rprege0d 10444 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( x  /  A
)  e.  RR  /\  0  <_  ( x  /  A ) ) )
18 flge0nn0 10995 . . . 4  |-  ( ( ( x  /  A
)  e.  RR  /\  0  <_  ( x  /  A ) )  -> 
( |_ `  (
x  /  A ) )  e.  NN0 )
19 nn0p1nn 10050 . . . 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 10310 . . 3  |-  NN  =  ( ZZ>= `  1 )
2220, 21syl6eleq 2406 . 2  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( |_ `  (
x  /  A ) )  +  1 )  e.  ( ZZ>= `  1
) )
2316rpred 10437 . . 3  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  e.  RR )
2410rpge0d 10441 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  0  <_  x )
2513adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <_  A )
264, 15, 2, 24, 25lediv2ad 10459 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  <_  ( x  / 
1 ) )
272recnd 8906 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  CC )
2827div1d 9573 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  1 )  =  x )
2926, 28breqtrd 4084 . . 3  |-  ( (
ph  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
x  /  A )  <_  x )
30 flword2 10990 . . 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 10862 . 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 1633    e. wcel 1701   A.wral 2577    u. cun 3184   ifcif 3599   class class class wbr 4060    e. cmpt 4114   ` cfv 5292  (class class class)co 5900   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    +oocpnf 8909    < clt 8912    <_ cle 8913    - cmin 9082    / cdiv 9468   NNcn 9791   NN0cn0 10012   ZZ>=cuz 10277   RR+crp 10401   (,)cioo 10703   ...cfz 10829   |_cfl 10971   abscabs 11766   sum_csu 12205   logclog 19965  Λcvma 20382  ψcchp 20383
This theorem is referenced by:  pntrlog2bndlem6  20785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-ioo 10707  df-fz 10830  df-fl 10972
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