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Theorem stoweidlem25 37879
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 3554 . . 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 10922 . . . . 5  |-  ( ph  ->  N  e.  NN0 )
6 stoweidlem25.3 . . . . . 6  |-  ( ph  ->  K  e.  NN )
76nnnn0d 10922 . . . . 5  |-  ( ph  ->  K  e.  NN0 )
82, 3, 5, 7stoweidlem12 37866 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `
 t ) ^ N ) ) ^
( K ^ N
) ) )
9 1red 9655 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
103fnvinran 37329 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
115adantr 467 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
1210, 11reexpcld 12430 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
139, 12resubcld 10044 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
146, 5nnexpcld 12434 . . . . . . 7  |-  ( ph  ->  ( K ^ N
)  e.  NN )
1514nnnn0d 10922 . . . . . 6  |-  ( ph  ->  ( K ^ N
)  e.  NN0 )
1615adantr 467 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( K ^ N )  e. 
NN0 )
1713, 16reexpcld 12430 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) )  e.  RR )
188, 17eqeltrd 2528 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  e.  RR )
191, 18sylan2 477 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  e.  RR )
206nnred 10621 . . . . . 6  |-  ( ph  ->  K  e.  RR )
21 stoweidlem25.4 . . . . . . 7  |-  ( ph  ->  D  e.  RR+ )
2221rpred 11338 . . . . . 6  |-  ( ph  ->  D  e.  RR )
2320, 22remulcld 9668 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  e.  RR )
2423, 5reexpcld 12430 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  e.  RR )
256nncnd 10622 . . . . . 6  |-  ( ph  ->  K  e.  CC )
266nnne0d 10651 . . . . . 6  |-  ( ph  ->  K  =/=  0 )
2721rpcnne0d 11347 . . . . . 6  |-  ( ph  ->  ( D  e.  CC  /\  D  =/=  0 ) )
28 mulne0 10251 . . . . . 6  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( K  x.  D
)  =/=  0 )
2925, 26, 27, 28syl21anc 1266 . . . . 5  |-  ( ph  ->  ( K  x.  D
)  =/=  0 )
3021rpcnd 11340 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
3125, 30mulcld 9660 . . . . . 6  |-  ( ph  ->  ( K  x.  D
)  e.  CC )
32 expne0 12300 . . . . . 6  |-  ( ( ( K  x.  D
)  e.  CC  /\  N  e.  NN )  ->  ( ( ( K  x.  D ) ^ N )  =/=  0  <->  ( K  x.  D )  =/=  0 ) )
3331, 4, 32syl2anc 666 . . . . 5  |-  ( ph  ->  ( ( ( K  x.  D ) ^ N )  =/=  0  <->  ( K  x.  D )  =/=  0 ) )
3429, 33mpbird 236 . . . 4  |-  ( ph  ->  ( ( K  x.  D ) ^ N
)  =/=  0 )
3524, 34rereccld 10431 . . 3  |-  ( ph  ->  ( 1  /  (
( K  x.  D
) ^ N ) )  e.  RR )
3635adantr 467 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  e.  RR )
37 stoweidlem25.9 . . . 4  |-  ( ph  ->  E  e.  RR+ )
3837rpred 11338 . . 3  |-  ( ph  ->  E  e.  RR )
3938adantr 467 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  E  e.  RR )
401, 8sylan2 477 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  =  ( ( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) ) )
414adantr 467 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  N  e.  NN )
426adantr 467 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  K  e.  NN )
4321adantr 467 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  e.  RR+ )
443adantr 467 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  P : T
--> RR )
451adantl 468 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  t  e.  T )
4644, 45ffvelrnd 6021 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  e.  RR )
47 0red 9641 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  e.  RR )
4822adantr 467 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  e.  RR )
4921rpgt0d 11341 . . . . . . 7  |-  ( ph  ->  0  <  D )
5049adantr 467 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  D )
51 stoweidlem25.8 . . . . . . 7  |-  ( ph  ->  A. t  e.  ( T  \  U ) D  <_  ( P `  t ) )
5251r19.21bi 2756 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  D  <_  ( P `  t ) )
5347, 48, 46, 50, 52ltletrd 9792 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <  ( P `  t ) )
5446, 53elrpd 11335 . . . 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 467 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_  1 ) )
57 rsp 2753 . . . . . 6  |-  ( A. t  e.  T  (
0  <_  ( P `  t )  /\  ( P `  t )  <_  1 )  ->  (
t  e.  T  -> 
( 0  <_  ( P `  t )  /\  ( P `  t
)  <_  1 ) ) )
5856, 45, 57sylc 62 . . . . 5  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_  1 ) )
5958simpld 461 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  0  <_  ( P `  t ) )
6058simprd 465 . . . 4  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( P `  t )  <_  1
)
6141, 42, 43, 54, 59, 60, 52stoweidlem1 37855 . . 3  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( (
1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) )  <_ 
( 1  /  (
( K  x.  D
) ^ N ) ) )
6240, 61eqbrtrd 4422 . 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 467 . 2  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  < 
E )
6519, 36, 39, 62, 64lelttrd 9790 1  |-  ( (
ph  /\  t  e.  ( T  \  U ) )  ->  ( Q `  t )  <  E
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736    \ cdif 3400   class class class wbr 4401    |-> cmpt 4460   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537    x. cmul 9541    < clt 9672    <_ cle 9673    - cmin 9857    / cdiv 10266   NNcn 10606   NN0cn0 10866   RR+crp 11299   ^cexp 12269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-seq 12211  df-exp 12270
This theorem is referenced by:  stoweidlem45  37900
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