Theorem stoweidlem25 37879

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 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 )
5	4	nnnn0d 10922	. . . . 5 |- ( ph -> N e. NN0 )
6		stoweidlem25.3	. . . . . 6 |- ( ph -> K e. NN )
7	6	nnnn0d 10922	. . . . 5 |- ( ph -> K e. NN0 )
8	2, 3, 5, 7	stoweidlem12 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 )
10	3	fnvinran 37329	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( P ` t ) e. RR )
11	5	adantr 467	. . . . . . 7 |- ( ( ph /\ t e. T ) -> N e. NN0 )
12	10, 11	reexpcld 12430	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( ( P ` t ) ^ N ) e. RR )
13	9, 12	resubcld 10044	. . . . 5 |- ( ( ph /\ t e. T ) -> ( 1 - ( ( P ` t ) ^ N ) ) e. RR )
14	6, 5	nnexpcld 12434	. . . . . . 7 |- ( ph -> ( K ^ N ) e. NN )
15	14	nnnn0d 10922	. . . . . 6 |- ( ph -> ( K ^ N ) e. NN0 )
16	15	adantr 467	. . . . 5 |- ( ( ph /\ t e. T ) -> ( K ^ N ) e. NN0 )
17	13, 16	reexpcld 12430	. . . 4 |- ( ( ph /\ t e. T ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) e. RR )
18	8, 17	eqeltrd 2528	. . 3 |- ( ( ph /\ t e. T ) -> ( Q ` t ) e. RR )
19	1, 18	sylan2 477	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) e. RR )
20	6	nnred 10621	. . . . . 6 |- ( ph -> K e. RR )
21		stoweidlem25.4	. . . . . . 7 |- ( ph -> D e. RR+ )
22	21	rpred 11338	. . . . . 6 |- ( ph -> D e. RR )
23	20, 22	remulcld 9668	. . . . 5 |- ( ph -> ( K x. D ) e. RR )
24	23, 5	reexpcld 12430	. . . 4 |- ( ph -> ( ( K x. D ) ^ N ) e. RR )
25	6	nncnd 10622	. . . . . 6 |- ( ph -> K e. CC )
26	6	nnne0d 10651	. . . . . 6 |- ( ph -> K =/= 0 )
27	21	rpcnne0d 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 )
29	25, 26, 27, 28	syl21anc 1266	. . . . 5 |- ( ph -> ( K x. D ) =/= 0 )
30	21	rpcnd 11340	. . . . . . 7 |- ( ph -> D e. CC )
31	25, 30	mulcld 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 ) )
33	31, 4, 32	syl2anc 666	. . . . 5 |- ( ph -> ( ( ( K x. D ) ^ N ) =/= 0 <-> ( K x. D ) =/= 0 ) )
34	29, 33	mpbird 236	. . . 4 |- ( ph -> ( ( K x. D ) ^ N ) =/= 0 )
35	24, 34	rereccld 10431	. . 3 |- ( ph -> ( 1 / ( ( K x. D ) ^ N ) ) e. RR )
36	35	adantr 467	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 1 / ( ( K x. D ) ^ N ) ) e. RR )
37		stoweidlem25.9	. . . 4 |- ( ph -> E e. RR+ )
38	37	rpred 11338	. . 3 |- ( ph -> E e. RR )
39	38	adantr 467	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> E e. RR )
40	1, 8	sylan2 477	. . 3 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) = ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) )
41	4	adantr 467	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> N e. NN )
42	6	adantr 467	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> K e. NN )
43	21	adantr 467	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> D e. RR+ )
44	3	adantr 467	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> P : T --> RR )
45	1	adantl 468	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> t e. T )
46	44, 45	ffvelrnd 6021	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> ( P ` t ) e. RR )
47		0red 9641	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 e. RR )
48	22	adantr 467	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> D e. RR )
49	21	rpgt0d 11341	. . . . . . 7 |- ( ph -> 0 < D )
50	49	adantr 467	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 < D )
51		stoweidlem25.8	. . . . . . 7 |- ( ph -> A. t e. ( T \ U ) D <_ ( P ` t ) )
52	51	r19.21bi 2756	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> D <_ ( P ` t ) )
53	47, 48, 46, 50, 52	ltletrd 9792	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 < ( P ` t ) )
54	46, 53	elrpd 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 ) )
56	55	adantr 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 ) ) )
58	56, 45, 57	sylc 62	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 0 <_ ( P ` t ) /\ ( P ` t ) <_ 1 ) )
59	58	simpld 461	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 <_ ( P ` t ) )
60	58	simprd 465	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> ( P ` t ) <_ 1 )
61	41, 42, 43, 54, 59, 60, 52	stoweidlem1 37855	. . 3 |- ( ( ph /\ t e. ( T \ U ) ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) <_ ( 1 / ( ( K x. D ) ^ N ) ) )
62	40, 61	eqbrtrd 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 )
64	63	adantr 467	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 1 / ( ( K x. D ) ^ N ) ) < E )
65	19, 36, 39, 62, 64	lelttrd 9790	1 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) < E )

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