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Theorem pntlema 24162
Description: Lemma for pnt 24180. 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 457 . . . . 5  |-  ( ph  ->  Y  e.  RR+ )
4 4nn 10736 . . . . . . 7  |-  4  e.  NN
5 nnrp 11274 . . . . . . 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 24160 . . . . . . . 8  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
1413simp1d 1009 . . . . . . 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 24161 . . . . . . . 8  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
2019simp1d 1009 . . . . . . 7  |-  ( ph  ->  E  e.  RR+ )
2114, 20rpmulcld 11320 . . . . . 6  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
22 rpdivcl 11288 . . . . . 6  |-  ( ( 4  e.  RR+  /\  ( L  x.  E )  e.  RR+ )  ->  (
4  /  ( L  x.  E ) )  e.  RR+ )
236, 21, 22sylancr 661 . . . . 5  |-  ( ph  ->  ( 4  /  ( L  x.  E )
)  e.  RR+ )
243, 23rpaddcld 11319 . . . 4  |-  ( ph  ->  ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+ )
25 2z 10937 . . . 4  |-  2  e.  ZZ
26 rpexpcl 12229 . . . 4  |-  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+  /\  2  e.  ZZ )  ->  (
( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  e.  RR+ )
2724, 25, 26sylancl 660 . . 3  |-  ( ph  ->  ( ( Y  +  ( 4  /  ( L  x.  E )
) ) ^ 2 )  e.  RR+ )
28 pntlem1.x . . . . . . 7  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
2928simpld 457 . . . . . 6  |-  ( ph  ->  X  e.  RR+ )
3019simp2d 1010 . . . . . . 7  |-  ( ph  ->  K  e.  RR+ )
31 rpexpcl 12229 . . . . . . 7  |-  ( ( K  e.  RR+  /\  2  e.  ZZ )  ->  ( K ^ 2 )  e.  RR+ )
3230, 25, 31sylancl 660 . . . . . 6  |-  ( ph  ->  ( K ^ 2 )  e.  RR+ )
3329, 32rpmulcld 11320 . . . . 5  |-  ( ph  ->  ( X  x.  ( K ^ 2 ) )  e.  RR+ )
34 4z 10939 . . . . 5  |-  4  e.  ZZ
35 rpexpcl 12229 . . . . 5  |-  ( ( ( X  x.  ( K ^ 2 ) )  e.  RR+  /\  4  e.  ZZ )  ->  (
( X  x.  ( K ^ 2 ) ) ^ 4 )  e.  RR+ )
3633, 34, 35sylancl 660 . . . 4  |-  ( ph  ->  ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  e.  RR+ )
37 3nn0 10854 . . . . . . . . . . 11  |-  3  e.  NN0
38 2nn 10734 . . . . . . . . . . 11  |-  2  e.  NN
3937, 38decnncl 11032 . . . . . . . . . 10  |- ; 3 2  e.  NN
40 nnrp 11274 . . . . . . . . . 10  |-  (; 3 2  e.  NN  -> ; 3
2  e.  RR+ )
4139, 40ax-mp 5 . . . . . . . . 9  |- ; 3 2  e.  RR+
42 rpmulcl 11287 . . . . . . . . 9  |-  ( (; 3
2  e.  RR+  /\  B  e.  RR+ )  ->  (; 3 2  x.  B )  e.  RR+ )
4341, 9, 42sylancr 661 . . . . . . . 8  |-  ( ph  ->  (; 3 2  x.  B
)  e.  RR+ )
4419simp3d 1011 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
4544simp3d 1011 . . . . . . . . 9  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
46 rpexpcl 12229 . . . . . . . . . . 11  |-  ( ( E  e.  RR+  /\  2  e.  ZZ )  ->  ( E ^ 2 )  e.  RR+ )
4720, 25, 46sylancl 660 . . . . . . . . . 10  |-  ( ph  ->  ( E ^ 2 )  e.  RR+ )
4814, 47rpmulcld 11320 . . . . . . . . 9  |-  ( ph  ->  ( L  x.  ( E ^ 2 ) )  e.  RR+ )
4945, 48rpmulcld 11320 . . . . . . . 8  |-  ( ph  ->  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) )  e.  RR+ )
5043, 49rpdivcld 11321 . . . . . . 7  |-  ( ph  ->  ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  e.  RR+ )
51 3nn 10735 . . . . . . . . . 10  |-  3  e.  NN
52 nnrp 11274 . . . . . . . . . 10  |-  ( 3  e.  NN  ->  3  e.  RR+ )
5351, 52ax-mp 5 . . . . . . . . 9  |-  3  e.  RR+
54 rpmulcl 11287 . . . . . . . . 9  |-  ( ( U  e.  RR+  /\  3  e.  RR+ )  ->  ( U  x.  3 )  e.  RR+ )
5515, 53, 54sylancl 660 . . . . . . . 8  |-  ( ph  ->  ( U  x.  3 )  e.  RR+ )
56 pntlem1.c . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
5755, 56rpaddcld 11319 . . . . . . 7  |-  ( ph  ->  ( ( U  x.  3 )  +  C
)  e.  RR+ )
5850, 57rpmulcld 11320 . . . . . 6  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR+ )
5958rpred 11304 . . . . 5  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR )
6059rpefcld 14049 . . . 4  |-  ( ph  ->  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) )  e.  RR+ )
6136, 60rpaddcld 11319 . . 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 11319 . 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 2494 1  |-  ( ph  ->  W  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4395    |-> cmpt 4453   ` cfv 5569  (class class class)co 6278   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    < clt 9658    <_ cle 9659    - cmin 9841    / cdiv 10247   NNcn 10576   2c2 10626   3c3 10627   4c4 10628   ZZcz 10905  ;cdc 11019   RR+crp 11265   (,)cioo 11582   ^cexp 12210   expce 14006  ψcchp 23747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-rp 11266  df-ioo 11586  df-ico 11588  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012
This theorem is referenced by:  pntlemb  24163  pntleme  24174
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