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Theorem stoweidlem25 27641
Description: This lemma proves that for n sufficiently large, qn( t ) < ε, for all  t in  T  \  U: see Lemma 1 [BrosowskiDeutsh] p. 91 (at the top of page 91).  Q is used to represent qn in the paper,  N to represent n in the paper,  K to represent k,  D to represent δ,  P to represent p, and  E to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem25.1  |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ ( K ^ N ) ) )
stoweidlem25.2  |-  ( ph  ->  N  e.  NN )
stoweidlem25.3  |-  ( ph  ->  K  e.  NN )
stoweidlem25.4  |-  ( ph  ->  D  e.  RR+ )
stoweidlem25.6  |-  ( ph  ->  P : T --> RR )
stoweidlem25.7  |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t
)  <_  1 ) )
stoweidlem25.8  |-  ( ph  ->  A. t  e.  ( T  \  U ) D  <_  ( P `  t ) )
stoweidlem25.9  |-  ( ph  ->  E  e.  RR+ )
stoweidlem25.11  |-  ( ph  ->  ( 1  /  (
( K  x.  D
) ^ N ) )  <  E )
Assertion
Ref Expression
stoweidlem25  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  <  E
)
Distinct variable group:    t, T
Allowed substitution hints:    ph( t)    D( t)    P( t)    Q( t)    U( t)    E( t)    K( t)    N( t)

Proof of Theorem stoweidlem25
StepHypRef Expression
1 eldifi 3429 . . 3  |-  ( t  e.  ( T  \  U )  ->  t  e.  T )
2 stoweidlem25.1 . . . . 5  |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ ( K ^ N ) ) )
3 stoweidlem25.6 . . . . 5  |-  ( ph  ->  P : T --> RR )
4 stoweidlem25.2 . . . . . 6  |-  ( ph  ->  N  e.  NN )
54nnnn0d 10230 . . . . 5  |-  ( ph  ->  N  e.  NN0 )
6 stoweidlem25.3 . . . . . 6  |-  ( ph  ->  K  e.  NN )
76nnnn0d 10230 . . . . 5  |-  ( ph  ->  K  e.  NN0 )
82, 3, 5, 7stoweidlem12 27628 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `
 t ) ^ N ) ) ^
( K ^ N
) ) )
9 1re 9046 . . . . . . 7  |-  1  e.  RR
109a1i 11 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
113fnvinran 27552 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
125adantr 452 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
1311, 12reexpcld 11495 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
1410, 13resubcld 9421 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
156, 5nnexpcld 11499 . . . . . . 7  |-  ( ph  ->  ( K ^ N
)  e.  NN )
1615nnnn0d 10230 . . . . . 6  |-  ( ph  ->  ( K ^ N
)  e.  NN0 )
1716adantr 452 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( K ^ N )  e. 
NN0 )
1814, 17reexpcld 11495 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) )  e.  RR )
198, 18eqeltrd 2478 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  e.  RR )
201, 19sylan2 461 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  e.  RR )
216nnred 9971 . . . . . 6  |-  ( ph  ->  K  e.  RR )
22 stoweidlem25.4 . . . . . . 7  |-  ( ph  ->  D  e.  RR+ )
2322rpred 10604 . . . . . 6  |-  ( ph  ->  D  e.  RR )
2421, 23remulcld 9072 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  e.  RR )
2524, 5reexpcld 11495 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  e.  RR )
266nncnd 9972 . . . . . 6  |-  ( ph  ->  K  e.  CC )
276nnne0d 10000 . . . . . 6  |-  ( ph  ->  K  =/=  0 )
2822rpcnne0d 10613 . . . . . 6  |-  ( ph  ->  ( D  e.  CC  /\  D  =/=  0 ) )
29 mulne0 9620 . . . . . 6  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( K  x.  D
)  =/=  0 )
3026, 27, 28, 29syl21anc 1183 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  =/=  0 )
3122rpcnd 10606 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
3226, 31mulcld 9064 . . . . . 6  |-  ( ph  ->  ( K  x.  D
)  e.  CC )
33 expne0 11366 . . . . . 6  |-  ( ( ( K  x.  D
)  e.  CC  /\  N  e.  NN )  ->  ( ( ( K  x.  D ) ^ N )  =/=  0  <->  ( K  x.  D )  =/=  0 ) )
3432, 4, 33syl2anc 643 . . . . 5  |-  ( ph  ->  ( ( ( K  x.  D ) ^ N )  =/=  0  <->  ( K  x.  D )  =/=  0 ) )
3530, 34mpbird 224 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  =/=  0 )
3625, 35rereccld 9797 . . 3  |-  ( ph  ->  ( 1  /  (
( K  x.  D
) ^ N ) )  e.  RR )
3736adantr 452 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  e.  RR )
38 stoweidlem25.9 . . . 4  |-  ( ph  ->  E  e.  RR+ )
3938rpred 10604 . . 3  |-  ( ph  ->  E  e.  RR )
4039adantr 452 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  E  e.  RR )
411, 8sylan2 461 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  =  ( ( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) ) )
424adantr 452 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  N  e.  NN )
436adantr 452 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  K  e.  NN )
4422adantr 452 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  e.  RR+ )
453adantr 452 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  P : T
--> RR )
461adantl 453 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  t  e.  T )
4745, 46ffvelrnd 5830 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  e.  RR )
48 0re 9047 . . . . . . 7  |-  0  e.  RR
4948a1i 11 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  e.  RR )
5023adantr 452 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  e.  RR )
5122rpgt0d 10607 . . . . . . 7  |-  ( ph  ->  0  <  D )
5251adantr 452 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  D )
53 stoweidlem25.8 . . . . . . 7  |-  ( ph  ->  A. t  e.  ( T  \  U ) D  <_  ( P `  t ) )
5453r19.21bi 2764 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  <_  ( P `  t ) )
5549, 50, 47, 52, 54ltletrd 9186 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  ( P `  t ) )
5647, 55elrpd 10602 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  e.  RR+ )
57 stoweidlem25.7 . . . . . . 7  |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t
)  <_  1 ) )
5857adantr 452 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_  1 ) )
59 rsp 2726 . . . . . 6  |-  ( A. t  e.  T  (
0  <_  ( P `  t )  /\  ( P `  t )  <_  1 )  ->  (
t  e.  T  -> 
( 0  <_  ( P `  t )  /\  ( P `  t
)  <_  1 ) ) )
6058, 46, 59sylc 58 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_  1 ) )
6160simpld 446 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <_  ( P `  t ) )
6260simprd 450 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  <_  1
)
6342, 43, 44, 56, 61, 62, 54stoweidlem1 27617 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( (
1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) )  <_ 
( 1  /  (
( K  x.  D
) ^ N ) ) )
6441, 63eqbrtrd 4192 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  <_  (
1  /  ( ( K  x.  D ) ^ N ) ) )
65 stoweidlem25.11 . . 3  |-  ( ph  ->  ( 1  /  (
( K  x.  D
) ^ N ) )  <  E )
6665adantr 452 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  < 
E )
6720, 37, 40, 64, 66lelttrd 9184 1  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  <  E
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666    \ cdif 3277   class class class wbr 4172    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   RR+crp 10568   ^cexp 11337
This theorem is referenced by:  stoweidlem45  27661
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338
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