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Theorem pntlema 20739
Description: Lemma for pnt 20757. 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 9874 . . . . . . 7  |-  4  e.  NN
5 nnrp 10358 . . . . . . 7  |-  ( 4  e.  NN  ->  4  e.  RR+ )
64, 5ax-mp 10 . . . . . 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 20737 . . . . . . . 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 20738 . . . . . . . 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 10401 . . . . . 6  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
22 rpdivcl 10371 . . . . . 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 10400 . . . 4  |-  ( ph  ->  ( Y  +  ( 4  /  ( L  x.  E ) ) )  e.  RR+ )
25 2z 10049 . . . 4  |-  2  e.  ZZ
26 rpexpcl 11116 . . . 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 970 . . . . . . 7  |-  ( ph  ->  K  e.  RR+ )
31 rpexpcl 11116 . . . . . . 7  |-  ( ( K  e.  RR+  /\  2  e.  ZZ )  ->  ( K ^ 2 )  e.  RR+ )
3230, 25, 31sylancl 645 . . . . . 6  |-  ( ph  ->  ( K ^ 2 )  e.  RR+ )
3329, 32rpmulcld 10401 . . . . 5  |-  ( ph  ->  ( X  x.  ( K ^ 2 ) )  e.  RR+ )
344nnzi 10042 . . . . 5  |-  4  e.  ZZ
35 rpexpcl 11116 . . . . 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 9978 . . . . . . . . . . 11  |-  3  e.  NN0
38 2nn 9872 . . . . . . . . . . 11  |-  2  e.  NN
3937, 38decnncl 10132 . . . . . . . . . 10  |- ; 3 2  e.  NN
40 nnrp 10358 . . . . . . . . . 10  |-  (; 3 2  e.  NN  -> ; 3
2  e.  RR+ )
4139, 40ax-mp 10 . . . . . . . . 9  |- ; 3 2  e.  RR+
42 rpmulcl 10370 . . . . . . . . 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 971 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
4544simp3d 971 . . . . . . . . 9  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
46 rpexpcl 11116 . . . . . . . . . . 11  |-  ( ( E  e.  RR+  /\  2  e.  ZZ )  ->  ( E ^ 2 )  e.  RR+ )
4720, 25, 46sylancl 645 . . . . . . . . . 10  |-  ( ph  ->  ( E ^ 2 )  e.  RR+ )
4814, 47rpmulcld 10401 . . . . . . . . 9  |-  ( ph  ->  ( L  x.  ( E ^ 2 ) )  e.  RR+ )
4945, 48rpmulcld 10401 . . . . . . . 8  |-  ( ph  ->  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) )  e.  RR+ )
5043, 49rpdivcld 10402 . . . . . . 7  |-  ( ph  ->  ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  e.  RR+ )
51 3nn 9873 . . . . . . . . . 10  |-  3  e.  NN
52 nnrp 10358 . . . . . . . . . 10  |-  ( 3  e.  NN  ->  3  e.  RR+ )
5351, 52ax-mp 10 . . . . . . . . 9  |-  3  e.  RR+
54 rpmulcl 10370 . . . . . . . . 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 10400 . . . . . . 7  |-  ( ph  ->  ( ( U  x.  3 )  +  C
)  e.  RR+ )
5850, 57rpmulcld 10401 . . . . . 6  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR+ )
5958rpred 10385 . . . . 5  |-  ( ph  ->  ( ( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) )  e.  RR )
6059rpefcld 12379 . . . 4  |-  ( ph  ->  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) )  e.  RR+ )
6136, 60rpaddcld 10400 . . 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 10400 . 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 2368 1  |-  ( ph  ->  W  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   0cc0 8732   1c1 8733    + caddc 8735    x. cmul 8737    < clt 8862    <_ cle 8863    - cmin 9032    / cdiv 9418   NNcn 9741   2c2 9790   3c3 9791   4c4 9792   ZZcz 10019  ;cdc 10119   RR+crp 10349   (,)cioo 10650   ^cexp 11098   expce 12337  ψcchp 20324
This theorem is referenced by:  pntlemb  20740  pntleme  20751
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-pm 6770  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-rp 10350  df-ioo 10654  df-ico 10656  df-fz 10777  df-fzo 10865  df-fl 10919  df-seq 11041  df-exp 11099  df-fac 11283  df-bc 11310  df-hash 11332  df-shft 11556  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-limsup 11939  df-clim 11956  df-rlim 11957  df-sum 12153  df-ef 12343
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