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Theorem pntlema 23502
Description: Lemma for pnt 23520. Closure for the constants used in the proof. The mammoth expression  W is a number large enough to satisfy all the lower bounds needed for  Z. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  Y is x2,  X is x1,  C is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and  W is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (Contributed by Mario Carneiro, 13-Apr-2016.)
Hypotheses
Ref Expression
pntlem1.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
pntlem1.a  |-  ( ph  ->  A  e.  RR+ )
pntlem1.b  |-  ( ph  ->  B  e.  RR+ )
pntlem1.l  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
pntlem1.d  |-  D  =  ( A  +  1 )
pntlem1.f  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
pntlem1.u  |-  ( ph  ->  U  e.  RR+ )
pntlem1.u2  |-  ( ph  ->  U  <_  A )
pntlem1.e  |-  E  =  ( U  /  D
)
pntlem1.k  |-  K  =  ( exp `  ( B  /  E ) )
pntlem1.y  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
pntlem1.x  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
pntlem1.c  |-  ( ph  ->  C  e.  RR+ )
pntlem1.w  |-  W  =  ( ( ( Y  +  ( 4  / 
( L  x.  E
) ) ) ^
2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) ) ) )
Assertion
Ref Expression
pntlema  |-  ( ph  ->  W  e.  RR+ )
Distinct variable group:    E, a
Allowed substitution hints:    ph( a)    A( a)    B( a)    C( a)    D( a)    R( a)    U( a)    F( a)    K( a)    L( a)    W( a)    X( a)    Y( a)

Proof of Theorem pntlema
StepHypRef Expression
1 pntlem1.w . 2  |-  W  =  ( ( ( Y  +  ( 4  / 
( L  x.  E
) ) ) ^
2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) ) ) )
2 pntlem1.y . . . . . 6  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
32simpld 459 . . . . 5  |-  ( ph  ->  Y  e.  RR+ )
4 4nn 10684 . . . . . . 7  |-  4  e.  NN
5 nnrp 11218 . . . . . . 7  |-  ( 4  e.  NN  ->  4  e.  RR+ )
64, 5ax-mp 5 . . . . . 6  |-  4  e.  RR+
7 pntlem1.r . . . . . . . . 9  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
8 pntlem1.a . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
9 pntlem1.b . . . . . . . . 9  |-  ( ph  ->  B  e.  RR+ )
10 pntlem1.l . . . . . . . . 9  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
11 pntlem1.d . . . . . . . . 9  |-  D  =  ( A  +  1 )
12 pntlem1.f . . . . . . . . 9  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
137, 8, 9, 10, 11, 12pntlemd 23500 . . . . . . . 8  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
1413simp1d 1003 . . . . . . 7  |-  ( ph  ->  L  e.  RR+ )
15 pntlem1.u . . . . . . . . 9  |-  ( ph  ->  U  e.  RR+ )
16 pntlem1.u2 . . . . . . . . 9  |-  ( ph  ->  U  <_  A )
17 pntlem1.e . . . . . . . . 9  |-  E  =  ( U  /  D
)
18 pntlem1.k . . . . . . . . 9  |-  K  =  ( exp `  ( B  /  E ) )
197, 8, 9, 10, 11, 12, 15, 16, 17, 18pntlemc 23501 . . . . . . . 8  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
2019simp1d 1003 . . . . . . 7  |-  ( ph  ->  E  e.  RR+ )
2114, 20rpmulcld 11261 . . . . . 6  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
22 rpdivcl 11231 . . . . . 6  |-  ( ( 4  e.  RR+  /\  ( L  x.  E )  e.  RR+ )  ->  (
4  /  ( L  x.  E ) )  e.  RR+ )
236, 21, 22sylancr 663 . . . . 5  |-  ( ph  ->  ( 4  /  ( L  x.  E )
)  e.  RR+ )
243, 23rpaddcld 11260 . . . 4  |-  ( ph  ->  ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+ )
25 2z 10885 . . . 4  |-  2  e.  ZZ
26 rpexpcl 12141 . . . 4  |-  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+  /\  2  e.  ZZ )  ->  (
( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  e.  RR+ )
2724, 25, 26sylancl 662 . . 3  |-  ( ph  ->  ( ( Y  +  ( 4  /  ( L  x.  E )
) ) ^ 2 )  e.  RR+ )
28 pntlem1.x . . . . . . 7  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
2928simpld 459 . . . . . 6  |-  ( ph  ->  X  e.  RR+ )
3019simp2d 1004 . . . . . . 7  |-  ( ph  ->  K  e.  RR+ )
31 rpexpcl 12141 . . . . . . 7  |-  ( ( K  e.  RR+  /\  2  e.  ZZ )  ->  ( K ^ 2 )  e.  RR+ )
3230, 25, 31sylancl 662 . . . . . 6  |-  ( ph  ->  ( K ^ 2 )  e.  RR+ )
3329, 32rpmulcld 11261 . . . . 5  |-  ( ph  ->  ( X  x.  ( K ^ 2 ) )  e.  RR+ )
344nnzi 10877 . . . . 5  |-  4  e.  ZZ
35 rpexpcl 12141 . . . . 5  |-  ( ( ( X  x.  ( K ^ 2 ) )  e.  RR+  /\  4  e.  ZZ )  ->  (
( X  x.  ( K ^ 2 ) ) ^ 4 )  e.  RR+ )
3633, 34, 35sylancl 662 . . . 4  |-  ( ph  ->  ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  e.  RR+ )
37 3nn0 10802 . . . . . . . . . . 11  |-  3  e.  NN0
38 2nn 10682 . . . . . . . . . . 11  |-  2  e.  NN
3937, 38decnncl 10978 . . . . . . . . . 10  |- ; 3 2  e.  NN
40 nnrp 11218 . . . . . . . . . 10  |-  (; 3 2  e.  NN  -> ; 3
2  e.  RR+ )
4139, 40ax-mp 5 . . . . . . . . 9  |- ; 3 2  e.  RR+
42 rpmulcl 11230 . . . . . . . . 9  |-  ( (; 3
2  e.  RR+  /\  B  e.  RR+ )  ->  (; 3 2  x.  B )  e.  RR+ )
4341, 9, 42sylancr 663 . . . . . . . 8  |-  ( ph  ->  (; 3 2  x.  B
)  e.  RR+ )
4419simp3d 1005 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
4544simp3d 1005 . . . . . . . . 9  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
46 rpexpcl 12141 . . . . . . . . . . 11  |-  ( ( E  e.  RR+  /\  2  e.  ZZ )  ->  ( E ^ 2 )  e.  RR+ )
4720, 25, 46sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( E ^ 2 )  e.  RR+ )
4814, 47rpmulcld 11261 . . . . . . . . 9  |-  ( ph  ->  ( L  x.  ( E ^ 2 ) )  e.  RR+ )
4945, 48rpmulcld 11261 . . . . . . . 8  |-  ( ph  ->  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) )  e.  RR+ )
5043, 49rpdivcld 11262 . . . . . . 7  |-  ( ph  ->  ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  e.  RR+ )
51 3nn 10683 . . . . . . . . . 10  |-  3  e.  NN
52 nnrp 11218 . . . . . . . . . 10  |-  ( 3  e.  NN  ->  3  e.  RR+ )
5351, 52ax-mp 5 . . . . . . . . 9  |-  3  e.  RR+
54 rpmulcl 11230 . . . . . . . . 9  |-  ( ( U  e.  RR+  /\  3  e.  RR+ )  ->  ( U  x.  3 )  e.  RR+ )
5515, 53, 54sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( U  x.  3 )  e.  RR+ )
56 pntlem1.c . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
5755, 56rpaddcld 11260 . . . . . . 7  |-  ( ph  ->  ( ( U  x.  3 )  +  C
)  e.  RR+ )
5850, 57rpmulcld 11261 . . . . . 6  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR+ )
5958rpred 11245 . . . . 5  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR )
6059rpefcld 13690 . . . 4  |-  ( ph  ->  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) )  e.  RR+ )
6136, 60rpaddcld 11260 . . 3  |-  ( ph  ->  ( ( ( X  x.  ( K ^
2 ) ) ^
4 )  +  ( exp `  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) ) )  e.  RR+ )
6227, 61rpaddcld 11260 . 2  |-  ( ph  ->  ( ( ( Y  +  ( 4  / 
( L  x.  E
) ) ) ^
2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) ) ) )  e.  RR+ )
631, 62syl5eqel 2552 1  |-  ( ph  ->  W  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9794    / cdiv 10195   NNcn 10525   2c2 10574   3c3 10575   4c4 10576   ZZcz 10853  ;cdc 10965   RR+crp 11209   (,)cioo 11518   ^cexp 12122   expce 13648  ψcchp 23087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-ioo 11522  df-ico 11524  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458  df-ef 13654
This theorem is referenced by:  pntlemb  23503  pntleme  23514
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