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Theorem stoweidlem25 27862
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 3458 . . 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 10312 . . . . 5  |-  ( ph  ->  N  e.  NN0 )
6 stoweidlem25.3 . . . . . 6  |-  ( ph  ->  K  e.  NN )
76nnnn0d 10312 . . . . 5  |-  ( ph  ->  K  e.  NN0 )
82, 3, 5, 7stoweidlem12 27849 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `
 t ) ^ N ) ) ^
( K ^ N
) ) )
9 1re 9128 . . . . . . 7  |-  1  e.  RR
109a1i 11 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
113fnvinran 27773 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
125adantr 453 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
1311, 12reexpcld 11578 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
1410, 13resubcld 9503 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
156, 5nnexpcld 11582 . . . . . . 7  |-  ( ph  ->  ( K ^ N
)  e.  NN )
1615nnnn0d 10312 . . . . . 6  |-  ( ph  ->  ( K ^ N
)  e.  NN0 )
1716adantr 453 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( K ^ N )  e. 
NN0 )
1814, 17reexpcld 11578 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) )  e.  RR )
198, 18eqeltrd 2517 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  e.  RR )
201, 19sylan2 462 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  e.  RR )
216nnred 10053 . . . . . 6  |-  ( ph  ->  K  e.  RR )
22 stoweidlem25.4 . . . . . . 7  |-  ( ph  ->  D  e.  RR+ )
2322rpred 10686 . . . . . 6  |-  ( ph  ->  D  e.  RR )
2421, 23remulcld 9154 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  e.  RR )
2524, 5reexpcld 11578 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  e.  RR )
266nncnd 10054 . . . . . 6  |-  ( ph  ->  K  e.  CC )
276nnne0d 10082 . . . . . 6  |-  ( ph  ->  K  =/=  0 )
2822rpcnne0d 10695 . . . . . 6  |-  ( ph  ->  ( D  e.  CC  /\  D  =/=  0 ) )
29 mulne0 9702 . . . . . 6  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( K  x.  D
)  =/=  0 )
3026, 27, 28, 29syl21anc 1184 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  =/=  0 )
3122rpcnd 10688 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
3226, 31mulcld 9146 . . . . . 6  |-  ( ph  ->  ( K  x.  D
)  e.  CC )
33 expne0 11449 . . . . . 6  |-  ( ( ( K  x.  D
)  e.  CC  /\  N  e.  NN )  ->  ( ( ( K  x.  D ) ^ N )  =/=  0  <->  ( K  x.  D )  =/=  0 ) )
3432, 4, 33syl2anc 644 . . . . 5  |-  ( ph  ->  ( ( ( K  x.  D ) ^ N )  =/=  0  <->  ( K  x.  D )  =/=  0 ) )
3530, 34mpbird 225 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  =/=  0 )
3625, 35rereccld 9879 . . 3  |-  ( ph  ->  ( 1  /  (
( K  x.  D
) ^ N ) )  e.  RR )
3736adantr 453 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  e.  RR )
38 stoweidlem25.9 . . . 4  |-  ( ph  ->  E  e.  RR+ )
3938rpred 10686 . . 3  |-  ( ph  ->  E  e.  RR )
4039adantr 453 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  E  e.  RR )
411, 8sylan2 462 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  =  ( ( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) ) )
424adantr 453 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  N  e.  NN )
436adantr 453 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  K  e.  NN )
4422adantr 453 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  e.  RR+ )
453adantr 453 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  P : T
--> RR )
461adantl 454 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  t  e.  T )
4745, 46ffvelrnd 5907 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  e.  RR )
48 0re 9129 . . . . . . 7  |-  0  e.  RR
4948a1i 11 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  e.  RR )
5023adantr 453 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  e.  RR )
5122rpgt0d 10689 . . . . . . 7  |-  ( ph  ->  0  <  D )
5251adantr 453 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  D )
53 stoweidlem25.8 . . . . . . 7  |-  ( ph  ->  A. t  e.  ( T  \  U ) D  <_  ( P `  t ) )
5453r19.21bi 2811 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  <_  ( P `  t ) )
5549, 50, 47, 52, 54ltletrd 9268 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  ( P `  t ) )
5647, 55elrpd 10684 . . . 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 453 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_  1 ) )
59 rsp 2773 . . . . . 6  |-  ( A. t  e.  T  (
0  <_  ( P `  t )  /\  ( P `  t )  <_  1 )  ->  (
t  e.  T  -> 
( 0  <_  ( P `  t )  /\  ( P `  t
)  <_  1 ) ) )
6058, 46, 59sylc 59 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_  1 ) )
6160simpld 447 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <_  ( P `  t ) )
6260simprd 451 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  <_  1
)
6342, 43, 44, 56, 61, 62, 54stoweidlem1 27838 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( (
1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) )  <_ 
( 1  /  (
( K  x.  D
) ^ N ) ) )
6441, 63eqbrtrd 4263 . 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 453 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  < 
E )
6720, 37, 40, 64, 66lelttrd 9266 1  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  <  E
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1654    e. wcel 1728    =/= wne 2606   A.wral 2712    \ cdif 3306   class class class wbr 4243    e. cmpt 4297   -->wf 5485   ` cfv 5489  (class class class)co 6117   CCcc 9026   RRcr 9027   0cc0 9028   1c1 9029    x. cmul 9033    < clt 9158    <_ cle 9159    - cmin 9329    / cdiv 9715   NNcn 10038   NN0cn0 10259   RR+crp 10650   ^cexp 11420
This theorem is referenced by:  stoweidlem45  27882
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 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-div 9716  df-nn 10039  df-2 10096  df-n0 10260  df-z 10321  df-uz 10527  df-rp 10651  df-seq 11362  df-exp 11421
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