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Theorem pntlema 21295
Description: Lemma for pnt 21313. 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 447 . . . . 5  |-  ( ph  ->  Y  e.  RR+ )
4 4nn 10140 . . . . . . 7  |-  4  e.  NN
5 nnrp 10626 . . . . . . 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 21293 . . . . . . . 8  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
1413simp1d 970 . . . . . . 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 21294 . . . . . . . 8  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
2019simp1d 970 . . . . . . 7  |-  ( ph  ->  E  e.  RR+ )
2114, 20rpmulcld 10669 . . . . . 6  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
22 rpdivcl 10639 . . . . . 6  |-  ( ( 4  e.  RR+  /\  ( L  x.  E )  e.  RR+ )  ->  (
4  /  ( L  x.  E ) )  e.  RR+ )
236, 21, 22sylancr 646 . . . . 5  |-  ( ph  ->  ( 4  /  ( L  x.  E )
)  e.  RR+ )
243, 23rpaddcld 10668 . . . 4  |-  ( ph  ->  ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+ )
25 2z 10317 . . . 4  |-  2  e.  ZZ
26 rpexpcl 11405 . . . 4  |-  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+  /\  2  e.  ZZ )  ->  (
( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  e.  RR+ )
2724, 25, 26sylancl 645 . . 3  |-  ( ph  ->  ( ( Y  +  ( 4  /  ( L  x.  E )
) ) ^ 2 )  e.  RR+ )
28 pntlem1.x . . . . . . 7  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
2928simpld 447 . . . . . 6  |-  ( ph  ->  X  e.  RR+ )
3019simp2d 971 . . . . . . 7  |-  ( ph  ->  K  e.  RR+ )
31 rpexpcl 11405 . . . . . . 7  |-  ( ( K  e.  RR+  /\  2  e.  ZZ )  ->  ( K ^ 2 )  e.  RR+ )
3230, 25, 31sylancl 645 . . . . . 6  |-  ( ph  ->  ( K ^ 2 )  e.  RR+ )
3329, 32rpmulcld 10669 . . . . 5  |-  ( ph  ->  ( X  x.  ( K ^ 2 ) )  e.  RR+ )
344nnzi 10310 . . . . 5  |-  4  e.  ZZ
35 rpexpcl 11405 . . . . 5  |-  ( ( ( X  x.  ( K ^ 2 ) )  e.  RR+  /\  4  e.  ZZ )  ->  (
( X  x.  ( K ^ 2 ) ) ^ 4 )  e.  RR+ )
3633, 34, 35sylancl 645 . . . 4  |-  ( ph  ->  ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  e.  RR+ )
37 3nn0 10244 . . . . . . . . . . 11  |-  3  e.  NN0
38 2nn 10138 . . . . . . . . . . 11  |-  2  e.  NN
3937, 38decnncl 10400 . . . . . . . . . 10  |- ; 3 2  e.  NN
40 nnrp 10626 . . . . . . . . . 10  |-  (; 3 2  e.  NN  -> ; 3
2  e.  RR+ )
4139, 40ax-mp 5 . . . . . . . . 9  |- ; 3 2  e.  RR+
42 rpmulcl 10638 . . . . . . . . 9  |-  ( (; 3
2  e.  RR+  /\  B  e.  RR+ )  ->  (; 3 2  x.  B )  e.  RR+ )
4341, 9, 42sylancr 646 . . . . . . . 8  |-  ( ph  ->  (; 3 2  x.  B
)  e.  RR+ )
4419simp3d 972 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
4544simp3d 972 . . . . . . . . 9  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
46 rpexpcl 11405 . . . . . . . . . . 11  |-  ( ( E  e.  RR+  /\  2  e.  ZZ )  ->  ( E ^ 2 )  e.  RR+ )
4720, 25, 46sylancl 645 . . . . . . . . . 10  |-  ( ph  ->  ( E ^ 2 )  e.  RR+ )
4814, 47rpmulcld 10669 . . . . . . . . 9  |-  ( ph  ->  ( L  x.  ( E ^ 2 ) )  e.  RR+ )
4945, 48rpmulcld 10669 . . . . . . . 8  |-  ( ph  ->  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) )  e.  RR+ )
5043, 49rpdivcld 10670 . . . . . . 7  |-  ( ph  ->  ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  e.  RR+ )
51 3nn 10139 . . . . . . . . . 10  |-  3  e.  NN
52 nnrp 10626 . . . . . . . . . 10  |-  ( 3  e.  NN  ->  3  e.  RR+ )
5351, 52ax-mp 5 . . . . . . . . 9  |-  3  e.  RR+
54 rpmulcl 10638 . . . . . . . . 9  |-  ( ( U  e.  RR+  /\  3  e.  RR+ )  ->  ( U  x.  3 )  e.  RR+ )
5515, 53, 54sylancl 645 . . . . . . . 8  |-  ( ph  ->  ( U  x.  3 )  e.  RR+ )
56 pntlem1.c . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
5755, 56rpaddcld 10668 . . . . . . 7  |-  ( ph  ->  ( ( U  x.  3 )  +  C
)  e.  RR+ )
5850, 57rpmulcld 10669 . . . . . 6  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR+ )
5958rpred 10653 . . . . 5  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR )
6059rpefcld 12711 . . . 4  |-  ( ph  ->  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) )  e.  RR+ )
6136, 60rpaddcld 10668 . . 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 10668 . 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 2522 1  |-  ( ph  ->  W  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000    < clt 9125    <_ cle 9126    - cmin 9296    / cdiv 9682   NNcn 10005   2c2 10054   3c3 10055   4c4 10056   ZZcz 10287  ;cdc 10387   RR+crp 10617   (,)cioo 10921   ^cexp 11387   expce 12669  ψcchp 20880
This theorem is referenced by:  pntlemb  21296  pntleme  21307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-rp 10618  df-ioo 10925  df-ico 10927  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675
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