Theorem stoweidlem25 31552

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 3626	. . 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 10853	. . . . 5 |- ( ph -> N e. NN0 )
6		stoweidlem25.3	. . . . . 6 |- ( ph -> K e. NN )
7	6	nnnn0d 10853	. . . . 5 |- ( ph -> K e. NN0 )
8	2, 3, 5, 7	stoweidlem12 31539	. . . 4 |- ( ( ph /\ t e. T ) -> ( Q ` t ) = ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) )
9		1red 9612	. . . . . 6 |- ( ( ph /\ t e. T ) -> 1 e. RR )
10	3	fnvinran 31194	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( P ` t ) e. RR )
11	5	adantr 465	. . . . . . 7 |- ( ( ph /\ t e. T ) -> N e. NN0 )
12	10, 11	reexpcld 12296	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( ( P ` t ) ^ N ) e. RR )
13	9, 12	resubcld 9988	. . . . 5 |- ( ( ph /\ t e. T ) -> ( 1 - ( ( P ` t ) ^ N ) ) e. RR )
14	6, 5	nnexpcld 12300	. . . . . . 7 |- ( ph -> ( K ^ N ) e. NN )
15	14	nnnn0d 10853	. . . . . 6 |- ( ph -> ( K ^ N ) e. NN0 )
16	15	adantr 465	. . . . 5 |- ( ( ph /\ t e. T ) -> ( K ^ N ) e. NN0 )
17	13, 16	reexpcld 12296	. . . 4 |- ( ( ph /\ t e. T ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) e. RR )
18	8, 17	eqeltrd 2555	. . 3 |- ( ( ph /\ t e. T ) -> ( Q ` t ) e. RR )
19	1, 18	sylan2 474	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) e. RR )
20	6	nnred 10552	. . . . . 6 |- ( ph -> K e. RR )
21		stoweidlem25.4	. . . . . . 7 |- ( ph -> D e. RR+ )
22	21	rpred 11257	. . . . . 6 |- ( ph -> D e. RR )
23	20, 22	remulcld 9625	. . . . 5 |- ( ph -> ( K x. D ) e. RR )
24	23, 5	reexpcld 12296	. . . 4 |- ( ph -> ( ( K x. D ) ^ N ) e. RR )
25	6	nncnd 10553	. . . . . 6 |- ( ph -> K e. CC )
26	6	nnne0d 10581	. . . . . 6 |- ( ph -> K =/= 0 )
27	21	rpcnne0d 11266	. . . . . 6 |- ( ph -> ( D e. CC /\ D =/= 0 ) )
28		mulne0 10192	. . . . . 6 |- ( ( ( K e. CC /\ K =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) -> ( K x. D ) =/= 0 )
29	25, 26, 27, 28	syl21anc 1227	. . . . 5 |- ( ph -> ( K x. D ) =/= 0 )
30	21	rpcnd 11259	. . . . . . 7 |- ( ph -> D e. CC )
31	25, 30	mulcld 9617	. . . . . 6 |- ( ph -> ( K x. D ) e. CC )
32		expne0 12166	. . . . . 6 |- ( ( ( K x. D ) e. CC /\ N e. NN ) -> ( ( ( K x. D ) ^ N ) =/= 0 <-> ( K x. D ) =/= 0 ) )
33	31, 4, 32	syl2anc 661	. . . . 5 |- ( ph -> ( ( ( K x. D ) ^ N ) =/= 0 <-> ( K x. D ) =/= 0 ) )
34	29, 33	mpbird 232	. . . 4 |- ( ph -> ( ( K x. D ) ^ N ) =/= 0 )
35	24, 34	rereccld 10372	. . 3 |- ( ph -> ( 1 / ( ( K x. D ) ^ N ) ) e. RR )
36	35	adantr 465	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 1 / ( ( K x. D ) ^ N ) ) e. RR )
37		stoweidlem25.9	. . . 4 |- ( ph -> E e. RR+ )
38	37	rpred 11257	. . 3 |- ( ph -> E e. RR )
39	38	adantr 465	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> E e. RR )
40	1, 8	sylan2 474	. . 3 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) = ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) )
41	4	adantr 465	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> N e. NN )
42	6	adantr 465	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> K e. NN )
43	21	adantr 465	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> D e. RR+ )
44	3	adantr 465	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> P : T --> RR )
45	1	adantl 466	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> t e. T )
46	44, 45	ffvelrnd 6023	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> ( P ` t ) e. RR )
47		0red 9598	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 e. RR )
48	22	adantr 465	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> D e. RR )
49	21	rpgt0d 11260	. . . . . . 7 |- ( ph -> 0 < D )
50	49	adantr 465	. . . . . 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 2833	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> D <_ ( P ` t ) )
53	47, 48, 46, 50, 52	ltletrd 9742	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 < ( P ` t ) )
54	46, 53	elrpd 11255	. . . 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 465	. . . . . 6 |- ( ( ph /\ t e. ( T \ U ) ) -> A. t e. T ( 0 <_ ( P ` t ) /\ ( P ` t ) <_ 1 ) )
57		rsp 2830	. . . . . 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 60	. . . . 5 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 0 <_ ( P ` t ) /\ ( P ` t ) <_ 1 ) )
59	58	simpld 459	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> 0 <_ ( P ` t ) )
60	58	simprd 463	. . . 4 |- ( ( ph /\ t e. ( T \ U ) ) -> ( P ` t ) <_ 1 )
61	41, 42, 43, 54, 59, 60, 52	stoweidlem1 31528	. . 3 |- ( ( ph /\ t e. ( T \ U ) ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ ( K ^ N ) ) <_ ( 1 / ( ( K x. D ) ^ N ) ) )
62	40, 61	eqbrtrd 4467	. 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 465	. 2 |- ( ( ph /\ t e. ( T \ U ) ) -> ( 1 / ( ( K x. D ) ^ N ) ) < E )
65	19, 36, 39, 62, 64	lelttrd 9740	1 |- ( ( ph /\ t e. ( T \ U ) ) -> ( Q ` t ) < E )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   \ cdif 3473   class class class wbr 4447   |-> cmpt 4505   -->wf 5584   ` cfv 5588  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494   x. cmul 9498   < clt 9629   <_ cle 9630   - cmin 9806   / cdiv 10207   NNcn 10537   NN0cn0 10796   RR+crp 11221   ^cexp 12135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-seq 12077  df-exp 12136

This theorem is referenced by:  stoweidlem45  31572