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Theorem stoweidlem25 27444
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 3414 . . 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 10208 . . . . 5  |-  ( ph  ->  N  e.  NN0 )
6 stoweidlem25.3 . . . . . 6  |-  ( ph  ->  K  e.  NN )
76nnnn0d 10208 . . . . 5  |-  ( ph  ->  K  e.  NN0 )
82, 3, 5, 7stoweidlem12 27431 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `
 t ) ^ N ) ) ^
( K ^ N
) ) )
9 1re 9025 . . . . . . 7  |-  1  e.  RR
109a1i 11 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
113fnvinran 27355 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
125adantr 452 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
1311, 12reexpcld 11469 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
1410, 13resubcld 9399 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
156, 5nnexpcld 11473 . . . . . . 7  |-  ( ph  ->  ( K ^ N
)  e.  NN )
1615nnnn0d 10208 . . . . . 6  |-  ( ph  ->  ( K ^ N
)  e.  NN0 )
1716adantr 452 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( K ^ N )  e. 
NN0 )
1814, 17reexpcld 11469 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) )  e.  RR )
198, 18eqeltrd 2463 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  e.  RR )
201, 19sylan2 461 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  e.  RR )
216nnred 9949 . . . . . 6  |-  ( ph  ->  K  e.  RR )
22 stoweidlem25.4 . . . . . . 7  |-  ( ph  ->  D  e.  RR+ )
2322rpred 10582 . . . . . 6  |-  ( ph  ->  D  e.  RR )
2421, 23remulcld 9051 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  e.  RR )
2524, 5reexpcld 11469 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  e.  RR )
266nncnd 9950 . . . . . 6  |-  ( ph  ->  K  e.  CC )
276nnne0d 9978 . . . . . 6  |-  ( ph  ->  K  =/=  0 )
2822rpcnne0d 10591 . . . . . 6  |-  ( ph  ->  ( D  e.  CC  /\  D  =/=  0 ) )
29 mulne0 9598 . . . . . 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 10584 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
3226, 31mulcld 9043 . . . . . 6  |-  ( ph  ->  ( K  x.  D
)  e.  CC )
33 expne0 11340 . . . . . 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 9775 . . 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 10582 . . 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 5812 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  e.  RR )
48 0re 9026 . . . . . . 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 10585 . . . . . . 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 2749 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  <_  ( P `  t ) )
5549, 50, 47, 52, 54ltletrd 9164 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  ( P `  t ) )
5647, 55elrpd 10580 . . . 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 2711 . . . . . 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 27420 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( (
1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) )  <_ 
( 1  /  (
( K  x.  D
) ^ N ) ) )
6441, 63eqbrtrd 4175 . 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 9162 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 1717    =/= wne 2552   A.wral 2651    \ cdif 3262   class class class wbr 4155    e. cmpt 4209   -->wf 5392   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    x. cmul 8930    < clt 9055    <_ cle 9056    - cmin 9225    / cdiv 9611   NNcn 9934   NN0cn0 10155   RR+crp 10546   ^cexp 11311
This theorem is referenced by:  stoweidlem45  27464
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-seq 11253  df-exp 11312
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