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Theorem pntlema 21251
Description: Lemma for pnt 21269. 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 446 . . . . 5  |-  ( ph  ->  Y  e.  RR+ )
4 4nn 10099 . . . . . . 7  |-  4  e.  NN
5 nnrp 10585 . . . . . . 7  |-  ( 4  e.  NN  ->  4  e.  RR+ )
64, 5ax-mp 8 . . . . . 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 21249 . . . . . . . 8  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
1413simp1d 969 . . . . . . 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 21250 . . . . . . . 8  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
2019simp1d 969 . . . . . . 7  |-  ( ph  ->  E  e.  RR+ )
2114, 20rpmulcld 10628 . . . . . 6  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
22 rpdivcl 10598 . . . . . 6  |-  ( ( 4  e.  RR+  /\  ( L  x.  E )  e.  RR+ )  ->  (
4  /  ( L  x.  E ) )  e.  RR+ )
236, 21, 22sylancr 645 . . . . 5  |-  ( ph  ->  ( 4  /  ( L  x.  E )
)  e.  RR+ )
243, 23rpaddcld 10627 . . . 4  |-  ( ph  ->  ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+ )
25 2z 10276 . . . 4  |-  2  e.  ZZ
26 rpexpcl 11363 . . . 4  |-  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+  /\  2  e.  ZZ )  ->  (
( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  e.  RR+ )
2724, 25, 26sylancl 644 . . 3  |-  ( ph  ->  ( ( Y  +  ( 4  /  ( L  x.  E )
) ) ^ 2 )  e.  RR+ )
28 pntlem1.x . . . . . . 7  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
2928simpld 446 . . . . . 6  |-  ( ph  ->  X  e.  RR+ )
3019simp2d 970 . . . . . . 7  |-  ( ph  ->  K  e.  RR+ )
31 rpexpcl 11363 . . . . . . 7  |-  ( ( K  e.  RR+  /\  2  e.  ZZ )  ->  ( K ^ 2 )  e.  RR+ )
3230, 25, 31sylancl 644 . . . . . 6  |-  ( ph  ->  ( K ^ 2 )  e.  RR+ )
3329, 32rpmulcld 10628 . . . . 5  |-  ( ph  ->  ( X  x.  ( K ^ 2 ) )  e.  RR+ )
344nnzi 10269 . . . . 5  |-  4  e.  ZZ
35 rpexpcl 11363 . . . . 5  |-  ( ( ( X  x.  ( K ^ 2 ) )  e.  RR+  /\  4  e.  ZZ )  ->  (
( X  x.  ( K ^ 2 ) ) ^ 4 )  e.  RR+ )
3633, 34, 35sylancl 644 . . . 4  |-  ( ph  ->  ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  e.  RR+ )
37 3nn0 10203 . . . . . . . . . . 11  |-  3  e.  NN0
38 2nn 10097 . . . . . . . . . . 11  |-  2  e.  NN
3937, 38decnncl 10359 . . . . . . . . . 10  |- ; 3 2  e.  NN
40 nnrp 10585 . . . . . . . . . 10  |-  (; 3 2  e.  NN  -> ; 3
2  e.  RR+ )
4139, 40ax-mp 8 . . . . . . . . 9  |- ; 3 2  e.  RR+
42 rpmulcl 10597 . . . . . . . . 9  |-  ( (; 3
2  e.  RR+  /\  B  e.  RR+ )  ->  (; 3 2  x.  B )  e.  RR+ )
4341, 9, 42sylancr 645 . . . . . . . 8  |-  ( ph  ->  (; 3 2  x.  B
)  e.  RR+ )
4419simp3d 971 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
4544simp3d 971 . . . . . . . . 9  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
46 rpexpcl 11363 . . . . . . . . . . 11  |-  ( ( E  e.  RR+  /\  2  e.  ZZ )  ->  ( E ^ 2 )  e.  RR+ )
4720, 25, 46sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( E ^ 2 )  e.  RR+ )
4814, 47rpmulcld 10628 . . . . . . . . 9  |-  ( ph  ->  ( L  x.  ( E ^ 2 ) )  e.  RR+ )
4945, 48rpmulcld 10628 . . . . . . . 8  |-  ( ph  ->  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) )  e.  RR+ )
5043, 49rpdivcld 10629 . . . . . . 7  |-  ( ph  ->  ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  e.  RR+ )
51 3nn 10098 . . . . . . . . . 10  |-  3  e.  NN
52 nnrp 10585 . . . . . . . . . 10  |-  ( 3  e.  NN  ->  3  e.  RR+ )
5351, 52ax-mp 8 . . . . . . . . 9  |-  3  e.  RR+
54 rpmulcl 10597 . . . . . . . . 9  |-  ( ( U  e.  RR+  /\  3  e.  RR+ )  ->  ( U  x.  3 )  e.  RR+ )
5515, 53, 54sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( U  x.  3 )  e.  RR+ )
56 pntlem1.c . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
5755, 56rpaddcld 10627 . . . . . . 7  |-  ( ph  ->  ( ( U  x.  3 )  +  C
)  e.  RR+ )
5850, 57rpmulcld 10628 . . . . . 6  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR+ )
5958rpred 10612 . . . . 5  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR )
6059rpefcld 12669 . . . 4  |-  ( ph  ->  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) )  e.  RR+ )
6136, 60rpaddcld 10627 . . 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 10627 . 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 2496 1  |-  ( ph  ->  W  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959    < clt 9084    <_ cle 9085    - cmin 9255    / cdiv 9641   NNcn 9964   2c2 10013   3c3 10014   4c4 10015   ZZcz 10246  ;cdc 10346   RR+crp 10576   (,)cioo 10880   ^cexp 11345   expce 12627  ψcchp 20836
This theorem is referenced by:  pntlemb  21252  pntleme  21263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-rp 10577  df-ioo 10884  df-ico 10886  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633
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