Theorem pntleme 25297

Description: Lemma for pnt 25303. Package up pntlemo 25296 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 25285	. 2 |- ( ph -> W e. RR+ )
16	2	adantr 481	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> A e. RR+ )
17	3	adantr 481	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> B e. RR+ )
18	4	adantr 481	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> L e. ( 0 (,) 1 ) )
19	7	adantr 481	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> U e. RR+ )
20	8	adantr 481	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> U <_ A )
21	11	adantr 481	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( Y e. RR+ /\ 1 <_ Y ) )
22	12	adantr 481	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( X e. RR+ /\ Y < X ) )
23	13	adantr 481	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> C e. RR+ )
24		simpr 477	. . . 4 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> v e. ( W [,) +oo ) )
25		eqid 2622	. . . 4 |- ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 )
26		eqid 2622	. . . 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 481	. . . 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 25284	. . . . . . . . 9 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )
30	29	simp2d 1074	. . . . . . . 8 |- ( ph -> K e. RR+ )
31	30	rpxrd 11873	. . . . . . 7 |- ( ph -> K e. RR* )
32		pnfxr 10092	. . . . . . . 8 |- +oo e. RR*
33	32	a1i 11	. . . . . . 7 |- ( ph -> +oo e. RR* )
34	30	rpred 11872	. . . . . . . 8 |- ( ph -> K e. RR )
35		ltpnf 11954	. . . . . . . 8 |- ( K e. RR -> K < +oo )
36	34, 35	syl 17	. . . . . . 7 |- ( ph -> K < +oo )
37		lbico1 12228	. . . . . . 7 |- ( ( K e. RR* /\ +oo e. RR* /\ K < +oo ) -> K e. ( K [,) +oo ) )
38	31, 33, 36, 37	syl3anc 1326	. . . . . 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 6657	. . . . . . . . . . . 12 |- ( k = K -> ( k x. y ) = ( K x. y ) )
41	40	breq2d 4665	. . . . . . . . . . 11 |- ( k = K -> ( ( ( 1 + ( L x. E ) ) x. z ) < ( k x. y ) <-> ( ( 1 + ( L x. E ) ) x. z ) < ( K x. y ) ) )
42	41	anbi2d 740	. . . . . . . . . 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 741	. . . . . . . . 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 3052	. . . . . . . 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 2986	. . . . . . 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 3307	. . . . . 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 693	. . . . 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 481	. . . 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 481	. . . 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 25296	. . 3 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
52	51	ralrimiva 2966	. 2 |- ( ph -> A. v e. ( W [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
53		oveq1 6657	. . . 4 |- ( w = W -> ( w [,) +oo ) = ( W [,) +oo ) )
54	53	raleqdv 3144	. . 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 3309	. 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 693	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   3c3 11071   4c4 11072  ;cdc 11493   RR+crp 11832   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   ...cfz 12326   |_cfl 12591   ^cexp 12860   abscabs 13974   sum_csu 14416   expce 14792   logclog 24301  ψcchp 24819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-em 24719  df-vma 24824  df-chp 24825

This theorem is referenced by:  pntlemp  25299