Theorem stoweidlem25 27649

Description: This lemma proves that for n sufficiently large, q_n( 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 q_n 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

Step	Hyp	Ref	Expression
1		eldifi 3437	. . 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 )
5	4	nnnn0d 10238	. . . . 5 |- ( ph -> N e. NN0 )
6		stoweidlem25.3	. . . . . 6 |- ( ph -> K e. NN )
7	6	nnnn0d 10238	. . . . 5 |- ( ph -> K e. NN0 )
8	2, 3, 5, 7	stoweidlem12 27636	. . . 4 |- ( ( ph /\ t e. T ) -> ( Q ` t ) = ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) )
9		1re 9054	. . . . . . 7 |- 1 e. RR
10	9	a1i 11	. . . . . 6 |- ( ( ph /\ t e. T ) -> 1 e. RR )
11	3	fnvinran 27560	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( P ` t ) e. RR )
12	5	adantr 452	. . . . . . 7 |- ( ( ph /\ t e. T ) -> N e. NN0 )
13	11, 12	reexpcld 11503	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( ( P ` t ) ^ N ) e. RR )
14	10, 13	resubcld 9429	. . . . 5 |- ( ( ph /\ t e. T ) -> ( 1 - ( ( P ` t ) ^ N ) ) e. RR )
15	6, 5	nnexpcld 11507	. . . . . . 7 |- ( ph -> ( K ^ N ) e. NN )
16	15	nnnn0d 10238	. . . . . 6 |- ( ph -> ( K ^ N ) e. NN0 )
17	16	adantr 452	. . . . 5 |- ( ( ph /\ t e. T ) -> ( K ^ N ) e. NN0 )
18	14, 17	reexpcld 11503	. . . 4 |- ( ( ph /\ t e. T ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) e. RR )
19	8, 18	eqeltrd 2486	. . 3 |- ( ( ph /\ t e. T ) -> ( Q ` t ) e. RR )
20	1, 19	sylan2 461	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) e. RR )
21	6	nnred 9979	. . . . . 6 |- ( ph -> K e. RR )
22		stoweidlem25.4	. . . . . . 7 |- ( ph -> D e. RR+ )
23	22	rpred 10612	. . . . . 6 |- ( ph -> D e. RR )
24	21, 23	remulcld 9080	. . . . 5 |- ( ph -> ( K x. D ) e. RR )
25	24, 5	reexpcld 11503	. . . 4 |- ( ph -> ( ( K x. D ) ^ N ) e. RR )
26	6	nncnd 9980	. . . . . 6 |- ( ph -> K e. CC )
27	6	nnne0d 10008	. . . . . 6 |- ( ph -> K =/= 0 )
28	22	rpcnne0d 10621	. . . . . 6 |- ( ph -> ( D e. CC /\ D =/= 0 ) )
29		mulne0 9628	. . . . . 6 |- ( ( ( K e. CC /\ K =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) -> ( K x. D ) =/= 0 )
30	26, 27, 28, 29	syl21anc 1183	. . . . 5 |- ( ph -> ( K x. D ) =/= 0 )
31	22	rpcnd 10614	. . . . . . 7 |- ( ph -> D e. CC )
32	26, 31	mulcld 9072	. . . . . 6 |- ( ph -> ( K x. D ) e. CC )
33		expne0 11374	. . . . . 6 |- ( ( ( K x. D ) e. CC /\ N e. NN ) -> ( ( ( K x. D ) ^ N ) =/= 0 <-> ( K x. D ) =/= 0 ) )
34	32, 4, 33	syl2anc 643	. . . . 5 |- ( ph -> ( ( ( K x. D ) ^ N ) =/= 0 <-> ( K x. D ) =/= 0 ) )
35	30, 34	mpbird 224	. . . 4 |- ( ph -> ( ( K x. D ) ^ N ) =/= 0 )
36	25, 35	rereccld 9805	. . 3 |- ( ph -> ( 1 / ( ( K x. D ) ^ N ) ) e. RR )
37	36	adantr 452	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 1 / ( ( K x. D ) ^ N ) ) e. RR )
38		stoweidlem25.9	. . . 4 |- ( ph -> E e. RR+ )
39	38	rpred 10612	. . 3 |- ( ph -> E e. RR )
40	39	adantr 452	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> E e. RR )
41	1, 8	sylan2 461	. . 3 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) = ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) )
42	4	adantr 452	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> N e. NN )
43	6	adantr 452	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> K e. NN )
44	22	adantr 452	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> D e. RR+ )
45	3	adantr 452	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> P : T --> RR )
46	1	adantl 453	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> t e. T )
47	45, 46	ffvelrnd 5838	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> ( P ` t ) e. RR )
48		0re 9055	. . . . . . 7 |- 0 e. RR
49	48	a1i 11	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 e. RR )
50	23	adantr 452	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> D e. RR )
51	22	rpgt0d 10615	. . . . . . 7 |- ( ph -> 0 < D )
52	51	adantr 452	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 < D )
53		stoweidlem25.8	. . . . . . 7 |- ( ph -> A. t e. ( T \ U ) D <_ ( P ` t ) )
54	53	r19.21bi 2772	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> D <_ ( P ` t ) )
55	49, 50, 47, 52, 54	ltletrd 9194	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 < ( P ` t ) )
56	47, 55	elrpd 10610	. . . 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 ) )
58	57	adantr 452	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> A. t e. T ( 0 <_ ( P ` t ) /\ ( P ` t ) <_ 1 ) )
59		rsp 2734	. . . . . 6 |- ( A. t e. T ( 0 <_ ( P ` t ) /\ ( P ` t ) <_ 1 ) -> ( t e. T -> ( 0 <_ ( P ` t ) /\ ( P ` t ) <_ 1 ) ) )
60	58, 46, 59	sylc 58	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 0 <_ ( P ` t ) /\ ( P ` t ) <_ 1 ) )
61	60	simpld 446	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 <_ ( P ` t ) )
62	60	simprd 450	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> ( P ` t ) <_ 1 )
63	42, 43, 44, 56, 61, 62, 54	stoweidlem1 27625	. . 3 |- ( ( ph /\ t e. ( T \ U ) ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) <_ ( 1 / ( ( K x. D ) ^ N ) ) )
64	41, 63	eqbrtrd 4200	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) <_ ( 1 / ( ( K x. D ) ^ N ) ) )
65		stoweidlem25.11	. . 3 |- ( ph -> ( 1 / ( ( K x. D ) ^ N ) ) < E )
66	65	adantr 452	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 1 / ( ( K x. D ) ^ N ) ) < E )
67	20, 37, 40, 64, 66	lelttrd 9192	1 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) < E )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2575   A.wral 2674   \ cdif 3285   class class class wbr 4180   e. cmpt 4234   -->wf 5417   ` cfv 5421  (class class class)co 6048   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955   x. cmul 8959   < clt 9084   <_ cle 9085   - cmin 9255   / cdiv 9641   NNcn 9964   NN0cn0 10185   RR+crp 10576   ^cexp 11345

This theorem is referenced by:  stoweidlem45  27669

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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031

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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-seq 11287  df-exp 11346