Theorem pntleme 23919

Description: Lemma for pnt 23925. Package up pntlemo 23918 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 23907	. 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 2457	. . . 4 |- ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 )
26		eqid 2457	. . . 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 23906	. . . . . . . . 9 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )
30	29	simp2d 1009	. . . . . . . 8 |- ( ph -> K e. RR+ )
31	30	rpxrd 11282	. . . . . . 7 |- ( ph -> K e. RR* )
32		pnfxr 11346	. . . . . . . 8 |- +oo e. RR*
33	32	a1i 11	. . . . . . 7 |- ( ph -> +oo e. RR* )
34	30	rpred 11281	. . . . . . . 8 |- ( ph -> K e. RR )
35		ltpnf 11356	. . . . . . . 8 |- ( K e. RR -> K < +oo )
36	34, 35	syl 16	. . . . . . 7 |- ( ph -> K < +oo )
37		lbico1 11604	. . . . . . 7 |- ( ( K e. RR* /\ +oo e. RR* /\ K < +oo ) -> K e. ( K [,) +oo ) )
38	31, 33, 36, 37	syl3anc 1228	. . . . . 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 6303	. . . . . . . . . . . 12 |- ( k = K -> ( k x. y ) = ( K x. y ) )
41	40	breq2d 4468	. . . . . . . . . . 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 2896	. . . . . . 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 3208	. . . . . 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 23918	. . 3 |- ( ( ph /\ v e. ( W [,) +oo ) ) -> ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
52	51	ralrimiva 2871	. 2 |- ( ph -> A. v e. ( W [,) +oo ) ( abs ` ( ( R ` v ) / v ) ) <_ ( U - ( F x. ( U ^ 3 ) ) ) )
53		oveq1 6303	. . . 4 |- ( w = W -> ( w [,) +oo ) = ( W [,) +oo ) )
54	53	raleqdv 3060	. . 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 3210	. 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 973   = wceq 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   class class class wbr 4456   |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   +oocpnf 9642   RR*cxr 9644   < clt 9645   <_ cle 9646   - cmin 9824   / cdiv 10227   2c2 10606   3c3 10607   4c4 10608  ;cdc 11000   RR+crp 11245   (,)cioo 11554   [,)cico 11556   [,]cicc 11557   ...cfz 11697   |_cfl 11930   ^cexp 12169   abscabs 13079   sum_csu 13520   expce 13809   logclog 23068  ψcchp 23492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-e 13816  df-sin 13817  df-cos 13818  df-pi 13820  df-dvds 13999  df-gcd 14157  df-prm 14230  df-pc 14373  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397  df-log 23070  df-em 23448  df-vma 23497  df-chp 23498

This theorem is referenced by:  pntlemp  23921