Theorem pntleme 23516

Description: Lemma for pnt 23522. Package up pntlemo 23515 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)

Hypotheses

Ref	Expression
pntlem1.r	|- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
pntlem1.a	|- ( ph -> A e. RR+ )
pntlem1.b	|- ( ph -> B e. RR+ )
pntlem1.l	|- ( ph -> L e. ( 0 (,) 1 ) )
pntlem1.d	|- D = ( A + 1 )
pntlem1.f	|- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
pntlem1.u	|- ( ph -> U e. RR+ )
pntlem1.u2	|- ( ph -> U <_ A )
pntlem1.e	|- E = ( U / D )
pntlem1.k	|- K = ( exp ` ( B / E ) )
pntlem1.y	|- ( ph -> ( Y e. RR+ /\ 1 <_ Y ) )
pntlem1.x	|- ( ph -> ( X e. RR+ /\ Y < X ) )
pntlem1.c	|- ( ph -> C e. RR+ )
pntlem1.w	|- W = ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) )
pntleme.U	|- ( ph -> A. z e. ( Y [,) +oo ) ( abs ` ( ( R ` z ) / z ) ) <_ U )
pntleme.K	|- ( ph -> A. k e. ( K [,) +oo ) A. y e. ( X (,) +oo ) E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) )
pntleme.C	|- ( ph -> A. z e. ( 1 (,) +oo ) ( ( ( ( abs ` ( R ` z ) ) x. ( log ` z ) ) - ( ( 2 / ( log ` z ) ) x. sum_ i e. ( 1 ... ( |_ ` ( z / Y ) ) ) ( ( abs ` ( R ` ( z / i ) ) ) x. ( log ` i ) ) ) ) / z ) <_ C )

Assertion

Ref	Expression
pntleme	|- ( ph -> E. w e. RR+ A. v e. ( w [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )

Distinct variable groups:   z, C   w, F   y, z   u, k, y, z, L   k, K, y, z   ph, v   i, k, u, v, w, y, z, R   w, U, z   v, W, w, z   k, X, y, z   i, Y, z   k, a, u, v, y, z, E

Allowed substitution hints:   ph( y, z, w, u, i, k, a)   A( y, z, w, v, u, i, k, a)   B( y, z, w, v, u, i, k, a)   C( y, w, v, u, i, k, a)   D( y, z, w, v, u, i, k, a)   R( a)   U( y, v, u, i, k, a)   E( w, i)   F( y, z, v, u, i, k, a)   K( w, v, u, i, a)   L( w, v, i, a)   W( y, u, i, k, a)   X( w, v, u, i, a)   Y( y, w, v, u, k, a)

Proof of Theorem pntleme

Step	Hyp	Ref	Expression
1		pntlem1.r	. . 3 |- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
2		pntlem1.a	. . 3 |- ( ph -> A e. RR+ )
3		pntlem1.b	. . 3 |- ( ph -> B e. RR+ )
4		pntlem1.l	. . 3 |- ( ph -> L e. ( 0 (,) 1 ) )
5		pntlem1.d	. . 3 |- D = ( A + 1 )
6		pntlem1.f	. . 3 |- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
7		pntlem1.u	. . 3 |- ( ph -> U e. RR+ )
8		pntlem1.u2	. . 3 |- ( ph -> U <_ A )
9		pntlem1.e	. . 3 |- E = ( U / D )
10		pntlem1.k	. . 3 |- K = ( exp ` ( B / E ) )
11		pntlem1.y	. . 3 |- ( ph -> ( Y e. RR+ /\ 1 <_ Y ) )
12		pntlem1.x	. . 3 |- ( ph -> ( X e. RR+ /\ Y < X ) )
13		pntlem1.c	. . 3 |- ( ph -> C e. RR+ )
14		pntlem1.w	. . 3 |- W = ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	pntlema 23504	. 2 |- ( ph -> W e. RR+ )
16	2	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> A e. RR+ )
17	3	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> B e. RR+ )
18	4	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> L e. ( 0 (,) 1 ) )
19	7	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> U e. RR+ )
20	8	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> U <_ A )
21	11	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( Y e. RR+ /\ 1 <_ Y ) )
22	12	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( X e. RR+ /\ Y < X ) )
23	13	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> C e. RR+ )
24		simpr 461	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> v e. ( W [,) +oo ) )
25		eqid 2462	. . . 4 |- ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 )
26		eqid 2462	. . . 4 |- ( |_ ` ( ( ( log ` v ) / ( log ` K ) ) / 2 ) ) = ( |_ ` ( ( ( log ` v ) / ( log ` K ) ) / 2 ) )
27		pntleme.U	. . . . 5 |- ( ph -> A. z e. ( Y [,) +oo ) ( abs ` ( ( R ` z ) / z ) ) <_ U )
28	27	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> A. z e. ( Y [,) +oo ) ( abs ` ( ( R ` z ) / z ) ) <_ U )
29	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 23503	. . . . . . . . 9 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )
30	29	simp2d 1004	. . . . . . . 8 |- ( ph -> K e. RR+ )
31	30	rpxrd 11248	. . . . . . 7 |- ( ph -> K e. RR* )
32		pnfxr 11312	. . . . . . . 8 |- +oo e. RR*
33	32	a1i 11	. . . . . . 7 |- ( ph -> +oo e. RR* )
34	30	rpred 11247	. . . . . . . 8 |- ( ph -> K e. RR )
35		ltpnf 11322	. . . . . . . 8 |- ( K e. RR -> K < +oo )
36	34, 35	syl 16	. . . . . . 7 |- ( ph -> K < +oo )
37		lbico1 11570	. . . . . . 7 |- ( ( K e. RR* /\ +oo e. RR* /\ K < +oo ) -> K e. ( K [,) +oo ) )
38	31, 33, 36, 37	syl3anc 1223	. . . . . 6 |- ( ph -> K e. ( K [,) +oo ) )
39		pntleme.K	. . . . . 6 |- ( ph -> A. k e. ( K [,) +oo ) A. y e. ( X (,) +oo ) E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) )
40		oveq1 6284	. . . . . . . . . . . 12 |- ( k = K -> ( k x. y ) = ( K x. y ) )
41	40	breq2d 4454	. . . . . . . . . . 11 |- ( k = K -> ( ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) <-> ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) )
42	41	anbi2d 703	. . . . . . . . . 10 |- ( k = K -> ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) ) <-> ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) ) )
43	42	anbi1d 704	. . . . . . . . 9 |- ( k = K -> ( ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) <-> ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) ) )
44	43	rexbidv 2968	. . . . . . . 8 |- ( k = K -> ( E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) <-> E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) ) )
45	44	ralbidv 2898	. . . . . . 7 |- ( k = K -> ( A. y e. ( X (,) +oo ) E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) <-> A. y e. ( X (,) +oo ) E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) ) )
46	45	rspcva 3207	. . . . . 6 |- ( ( K e. ( K [,) +oo ) /\ A. k e. ( K [,) +oo ) A. y e. ( X (,) +oo ) E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) ) -> A. y e. ( X (,) +oo ) E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) )
47	38, 39, 46	syl2anc 661	. . . . 5 |- ( ph -> A. y e. ( X (,) +oo ) E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) )
48	47	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> A. y e. ( X (,) +oo ) E. z e. RR+ ( ( y < z /\ ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) /\ A. u e. ( z [,] ( ( 1 + ( L x. E ) ) x. z ) ) ( abs ` ( ( R ` u ) / u ) ) <_ E ) )
49		pntleme.C	. . . . 5 |- ( ph -> A. z e. ( 1 (,) +oo ) ( ( ( ( abs ` ( R ` z ) ) x. ( log ` z ) ) - ( ( 2 / ( log ` z ) ) x. sum_ i e. ( 1 ... ( |_ ` ( z / Y ) ) ) ( ( abs ` ( R ` ( z / i ) ) ) x. ( log ` i ) ) ) ) / z ) <_ C )
50	49	adantr 465	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> A. z e. ( 1 (,) +oo ) ( ( ( ( abs ` ( R ` z ) ) x. ( log ` z ) ) - ( ( 2 / ( log ` z ) ) x. sum_ i e. ( 1 ... ( |_ ` ( z / Y ) ) ) ( ( abs ` ( R ` ( z / i ) ) ) x. ( log ` i ) ) ) ) / z ) <_ C )
51	1, 16, 17, 18, 5, 6, 19, 20, 9, 10, 21, 22, 23, 14, 24, 25, 26, 28, 48, 50	pntlemo 23515	. . 3 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
52	51	ralrimiva 2873	. 2 |- ( ph -> A. v e. ( W [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
53		oveq1 6284	. . . 4 |- ( w = W -> ( w [,) +oo ) = ( W [,) +oo ) )
54	53	raleqdv 3059	. . 3 |- ( w = W -> ( A. v e. ( w [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) <-> A. v e. ( W [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) ) )
55	54	rspcev 3209	. 2 |- ( ( W e. RR+ /\ A. v e. ( W [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) ) -> E. w e. RR+ A. v e. ( w [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
56	15, 52, 55	syl2anc 661	1 |- ( ph -> E. w e. RR+ A. v e. ( w [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   A.wral 2809   E.wrex 2810   class class class wbr 4442   |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   RRcr 9482   0cc0 9483   1c1 9484   + caddc 9486   x. cmul 9488   +oocpnf 9616   RR*cxr 9618   < clt 9619   <_ cle 9620   - cmin 9796   / cdiv 10197   2c2 10576   3c3 10577   4c4 10578  ;cdc 10967   RR+crp 11211   (,)cioo 11520   [,)cico 11522   [,]cicc 11523   ...cfz 11663   |_cfl 11886   ^cexp 12124   abscabs 13019   sum_csu 13459   expce 13650   logclog 22665  ψcchp 23089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-e 13657  df-sin 13658  df-cos 13659  df-pi 13661  df-dvds 13839  df-gcd 13995  df-prm 14068  df-pc 14211  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001  df-log 22667  df-em 23045  df-vma 23094  df-chp 23095

This theorem is referenced by:  pntlemp  23518