Theorem stoweidlem25 27862

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 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 )
5	4	nnnn0d 10312	. . . . 5 |- ( ph -> N e. NN0 )
6		stoweidlem25.3	. . . . . 6 |- ( ph -> K e. NN )
7	6	nnnn0d 10312	. . . . 5 |- ( ph -> K e. NN0 )
8	2, 3, 5, 7	stoweidlem12 27849	. . . 4 |- ( ( ph /\ t e. T ) -> ( Q ` t ) = ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) )
9		1re 9128	. . . . . . 7 |- 1 e. RR
10	9	a1i 11	. . . . . 6 |- ( ( ph /\ t e. T ) -> 1 e. RR )
11	3	fnvinran 27773	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( P ` t ) e. RR )
12	5	adantr 453	. . . . . . 7 |- ( ( ph /\ t e. T ) -> N e. NN0 )
13	11, 12	reexpcld 11578	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( ( P ` t ) ^ N ) e. RR )
14	10, 13	resubcld 9503	. . . . 5 |- ( ( ph /\ t e. T ) -> ( 1 - ( ( P ` t ) ^ N ) ) e. RR )
15	6, 5	nnexpcld 11582	. . . . . . 7 |- ( ph -> ( K ^ N ) e. NN )
16	15	nnnn0d 10312	. . . . . 6 |- ( ph -> ( K ^ N ) e. NN0 )
17	16	adantr 453	. . . . 5 |- ( ( ph /\ t e. T ) -> ( K ^ N ) e. NN0 )
18	14, 17	reexpcld 11578	. . . 4 |- ( ( ph /\ t e. T ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) e. RR )
19	8, 18	eqeltrd 2517	. . 3 |- ( ( ph /\ t e. T ) -> ( Q ` t ) e. RR )
20	1, 19	sylan2 462	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) e. RR )
21	6	nnred 10053	. . . . . 6 |- ( ph -> K e. RR )
22		stoweidlem25.4	. . . . . . 7 |- ( ph -> D e. RR+ )
23	22	rpred 10686	. . . . . 6 |- ( ph -> D e. RR )
24	21, 23	remulcld 9154	. . . . 5 |- ( ph -> ( K x. D ) e. RR )
25	24, 5	reexpcld 11578	. . . 4 |- ( ph -> ( ( K x. D ) ^ N ) e. RR )
26	6	nncnd 10054	. . . . . 6 |- ( ph -> K e. CC )
27	6	nnne0d 10082	. . . . . 6 |- ( ph -> K =/= 0 )
28	22	rpcnne0d 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 )
30	26, 27, 28, 29	syl21anc 1184	. . . . 5 |- ( ph -> ( K x. D ) =/= 0 )
31	22	rpcnd 10688	. . . . . . 7 |- ( ph -> D e. CC )
32	26, 31	mulcld 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 ) )
34	32, 4, 33	syl2anc 644	. . . . 5 |- ( ph -> ( ( ( K x. D ) ^ N ) =/= 0 <-> ( K x. D ) =/= 0 ) )
35	30, 34	mpbird 225	. . . 4 |- ( ph -> ( ( K x. D ) ^ N ) =/= 0 )
36	25, 35	rereccld 9879	. . 3 |- ( ph -> ( 1 / ( ( K x. D ) ^ N ) ) e. RR )
37	36	adantr 453	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 1 / ( ( K x. D ) ^ N ) ) e. RR )
38		stoweidlem25.9	. . . 4 |- ( ph -> E e. RR+ )
39	38	rpred 10686	. . 3 |- ( ph -> E e. RR )
40	39	adantr 453	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> E e. RR )
41	1, 8	sylan2 462	. . 3 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) = ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) )
42	4	adantr 453	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> N e. NN )
43	6	adantr 453	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> K e. NN )
44	22	adantr 453	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> D e. RR+ )
45	3	adantr 453	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> P : T --> RR )
46	1	adantl 454	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> t e. T )
47	45, 46	ffvelrnd 5907	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> ( P ` t ) e. RR )
48		0re 9129	. . . . . . 7 |- 0 e. RR
49	48	a1i 11	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 e. RR )
50	23	adantr 453	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> D e. RR )
51	22	rpgt0d 10689	. . . . . . 7 |- ( ph -> 0 < D )
52	51	adantr 453	. . . . . 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 2811	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> D <_ ( P ` t ) )
55	49, 50, 47, 52, 54	ltletrd 9268	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 < ( P ` t ) )
56	47, 55	elrpd 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 ) )
58	57	adantr 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 ) ) )
60	58, 46, 59	sylc 59	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 0 <_ ( P ` t ) /\ ( P ` t ) <_ 1 ) )
61	60	simpld 447	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 <_ ( P ` t ) )
62	60	simprd 451	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> ( P ` t ) <_ 1 )
63	42, 43, 44, 56, 61, 62, 54	stoweidlem1 27838	. . 3 |- ( ( ph /\ t e. ( T \ U ) ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) <_ ( 1 / ( ( K x. D ) ^ N ) ) )
64	41, 63	eqbrtrd 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 )
66	65	adantr 453	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 1 / ( ( K x. D ) ^ N ) ) < E )
67	20, 37, 40, 64, 66	lelttrd 9266	1 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) < E )

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