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Theorem stoweidlem25 31968
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 3622 . . 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 10873 . . . . 5  |-  ( ph  ->  N  e.  NN0 )
6 stoweidlem25.3 . . . . . 6  |-  ( ph  ->  K  e.  NN )
76nnnn0d 10873 . . . . 5  |-  ( ph  ->  K  e.  NN0 )
82, 3, 5, 7stoweidlem12 31955 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `
 t ) ^ N ) ) ^
( K ^ N
) ) )
9 1red 9628 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
103fnvinran 31550 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
115adantr 465 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
1210, 11reexpcld 12329 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
139, 12resubcld 10008 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
146, 5nnexpcld 12333 . . . . . . 7  |-  ( ph  ->  ( K ^ N
)  e.  NN )
1514nnnn0d 10873 . . . . . 6  |-  ( ph  ->  ( K ^ N
)  e.  NN0 )
1615adantr 465 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( K ^ N )  e. 
NN0 )
1713, 16reexpcld 12329 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) )  e.  RR )
188, 17eqeltrd 2545 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  e.  RR )
191, 18sylan2 474 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  e.  RR )
206nnred 10571 . . . . . 6  |-  ( ph  ->  K  e.  RR )
21 stoweidlem25.4 . . . . . . 7  |-  ( ph  ->  D  e.  RR+ )
2221rpred 11281 . . . . . 6  |-  ( ph  ->  D  e.  RR )
2320, 22remulcld 9641 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  e.  RR )
2423, 5reexpcld 12329 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  e.  RR )
256nncnd 10572 . . . . . 6  |-  ( ph  ->  K  e.  CC )
266nnne0d 10601 . . . . . 6  |-  ( ph  ->  K  =/=  0 )
2721rpcnne0d 11290 . . . . . 6  |-  ( ph  ->  ( D  e.  CC  /\  D  =/=  0 ) )
28 mulne0 10212 . . . . . 6  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( K  x.  D
)  =/=  0 )
2925, 26, 27, 28syl21anc 1227 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  =/=  0 )
3021rpcnd 11283 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
3125, 30mulcld 9633 . . . . . 6  |-  ( ph  ->  ( K  x.  D
)  e.  CC )
32 expne0 12199 . . . . . 6  |-  ( ( ( K  x.  D
)  e.  CC  /\  N  e.  NN )  ->  ( ( ( K  x.  D ) ^ N )  =/=  0  <->  ( K  x.  D )  =/=  0 ) )
3331, 4, 32syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( ( K  x.  D ) ^ N )  =/=  0  <->  ( K  x.  D )  =/=  0 ) )
3429, 33mpbird 232 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  =/=  0 )
3524, 34rereccld 10392 . . 3  |-  ( ph  ->  ( 1  /  (
( K  x.  D
) ^ N ) )  e.  RR )
3635adantr 465 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  e.  RR )
37 stoweidlem25.9 . . . 4  |-  ( ph  ->  E  e.  RR+ )
3837rpred 11281 . . 3  |-  ( ph  ->  E  e.  RR )
3938adantr 465 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  E  e.  RR )
401, 8sylan2 474 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  =  ( ( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) ) )
414adantr 465 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  N  e.  NN )
426adantr 465 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  K  e.  NN )
4321adantr 465 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  e.  RR+ )
443adantr 465 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  P : T
--> RR )
451adantl 466 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  t  e.  T )
4644, 45ffvelrnd 6033 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  e.  RR )
47 0red 9614 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  e.  RR )
4822adantr 465 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  e.  RR )
4921rpgt0d 11284 . . . . . . 7  |-  ( ph  ->  0  <  D )
5049adantr 465 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  D )
51 stoweidlem25.8 . . . . . . 7  |-  ( ph  ->  A. t  e.  ( T  \  U ) D  <_  ( P `  t ) )
5251r19.21bi 2826 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  <_  ( P `  t ) )
5347, 48, 46, 50, 52ltletrd 9759 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  ( P `  t ) )
5446, 53elrpd 11279 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  e.  RR+ )
55 stoweidlem25.7 . . . . . . 7  |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t
)  <_  1 ) )
5655adantr 465 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_  1 ) )
57 rsp 2823 . . . . . 6  |-  ( A. t  e.  T  (
0  <_  ( P `  t )  /\  ( P `  t )  <_  1 )  ->  (
t  e.  T  -> 
( 0  <_  ( P `  t )  /\  ( P `  t
)  <_  1 ) ) )
5856, 45, 57sylc 60 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_  1 ) )
5958simpld 459 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <_  ( P `  t ) )
6058simprd 463 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  <_  1
)
6141, 42, 43, 54, 59, 60, 52stoweidlem1 31944 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( (
1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) )  <_ 
( 1  /  (
( K  x.  D
) ^ N ) ) )
6240, 61eqbrtrd 4476 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  <_  (
1  /  ( ( K  x.  D ) ^ N ) ) )
63 stoweidlem25.11 . . 3  |-  ( ph  ->  ( 1  /  (
( K  x.  D
) ^ N ) )  <  E )
6463adantr 465 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  < 
E )
6519, 36, 39, 62, 64lelttrd 9757 1  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  <  E
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    \ cdif 3468   class class class wbr 4456    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   NN0cn0 10816   RR+crp 11245   ^cexp 12168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12110  df-exp 12169
This theorem is referenced by:  stoweidlem45  31988
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