Theorem pntleme 24458

Description: Lemma for pnt 24464. Package up pntlemo 24457 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 24446	. 2 |- ( ph -> W e. RR+ )
16	2	adantr 467	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> A e. RR+ )
17	3	adantr 467	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> B e. RR+ )
18	4	adantr 467	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> L e. ( 0 (,) 1 ) )
19	7	adantr 467	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> U e. RR+ )
20	8	adantr 467	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> U <_ A )
21	11	adantr 467	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( Y e. RR+ /\ 1 <_ Y ) )
22	12	adantr 467	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( X e. RR+ /\ Y < X ) )
23	13	adantr 467	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> C e. RR+ )
24		simpr 463	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> v e. ( W [,) +oo ) )
25		eqid 2453	. . . 4 |- ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 )
26		eqid 2453	. . . 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 467	. . . 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 24445	. . . . . . . . 9 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )
30	29	simp2d 1022	. . . . . . . 8 |- ( ph -> K e. RR+ )
31	30	rpxrd 11349	. . . . . . 7 |- ( ph -> K e. RR* )
32		pnfxr 11419	. . . . . . . 8 |- +oo e. RR*
33	32	a1i 11	. . . . . . 7 |- ( ph -> +oo e. RR* )
34	30	rpred 11348	. . . . . . . 8 |- ( ph -> K e. RR )
35		ltpnf 11429	. . . . . . . 8 |- ( K e. RR -> K < +oo )
36	34, 35	syl 17	. . . . . . 7 |- ( ph -> K < +oo )
37		lbico1 11696	. . . . . . 7 |- ( ( K e. RR* /\ +oo e. RR* /\ K < +oo ) -> K e. ( K [,) +oo ) )
38	31, 33, 36, 37	syl3anc 1269	. . . . . 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 6302	. . . . . . . . . . . 12 |- ( k = K -> ( k x. y ) = ( K x. y ) )
41	40	breq2d 4417	. . . . . . . . . . 11 |- ( k = K -> ( ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) <-> ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) )
42	41	anbi2d 711	. . . . . . . . . 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 712	. . . . . . . . 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 2903	. . . . . . . 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 2829	. . . . . . 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 3150	. . . . . 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 667	. . . . 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 467	. . . 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 467	. . . 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 24457	. . 3 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
52	51	ralrimiva 2804	. 2 |- ( ph -> A. v e. ( W [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
53		oveq1 6302	. . . 4 |- ( w = W -> ( w [,) +oo ) = ( W [,) +oo ) )
54	53	raleqdv 2995	. . 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 3152	. 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 667	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 371   /\ w3a 986   = wceq 1446   e. wcel 1889   A.wral 2739   E.wrex 2740   class class class wbr 4405   |-> cmpt 4464   ` cfv 5585  (class class class)co 6295   RRcr 9543   0cc0 9544   1c1 9545   + caddc 9547   x. cmul 9549   +oocpnf 9677   RR*cxr 9679   < clt 9680   <_ cle 9681   - cmin 9865   / cdiv 10276   2c2 10666   3c3 10667   4c4 10668  ;cdc 11058   RR+crp 11309   (,)cioo 11642   [,)cico 11644   [,]cicc 11645   ...cfz 11791   |_cfl 12033   ^cexp 12279   abscabs 13309   sum_csu 13764   expce 14126   logclog 23516  ψcchp 24031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623  ax-mulf 9624

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ioc 11647  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-shft 13142  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-limsup 13538  df-clim 13564  df-rlim 13565  df-sum 13765  df-ef 14133  df-e 14134  df-sin 14135  df-cos 14136  df-pi 14138  df-dvds 14318  df-gcd 14481  df-prm 14635  df-pc 14799  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-fbas 18979  df-fg 18980  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-ntr 20047  df-cls 20048  df-nei 20126  df-lp 20164  df-perf 20165  df-cn 20255  df-cnp 20256  df-haus 20343  df-tx 20589  df-hmeo 20782  df-fil 20873  df-fm 20965  df-flim 20966  df-flf 20967  df-xms 21347  df-ms 21348  df-tms 21349  df-cncf 21922  df-limc 22833  df-dv 22834  df-log 23518  df-em 23930  df-vma 24036  df-chp 24037

This theorem is referenced by:  pntlemp  24460