pntrlog2bnd - Metamath Proof Explorer

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Theorem pntrlog2bnd 21231

Description: A bound on

. Equation 10.6.15 of [Shapiro], p. 431. (Contributed by Mario Carneiro, 1-Jun-2016.)

**Hypothesis**
Ref	Expression
pntpbnd.r	ψ

**Assertion**
Ref	Expression
pntrlog2bnd	$/\$

Distinct variable groups:

Allowed substitution hint:

(

)

Proof of Theorem pntrlog2bnd Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioossre 10928	. . 3
2	1	a1i 11	. 2 $/\$
3		1re 9046	. . 3
4	3	a1i 11	. 2 $/\$
5	2	sselda 3308	. . . . . . . . 9 $/\$ $/\$
6		1rp 10572	. . . . . . . . . 10
7	6	a1i 11	. . . . . . . . 9 $/\$ $/\$
8	3	a1i 11	. . . . . . . . . 10 $/\$ $/\$
9		eliooord 10926	. . . . . . . . . . . 12 $/\$
10	9	adantl 453	. . . . . . . . . . 11 $/\$ $/\$ $/\$
11	10	simpld 446	. . . . . . . . . 10 $/\$ $/\$
12	8, 5, 11	ltled 9177	. . . . . . . . 9 $/\$ $/\$
13	5, 7, 12	rpgecld 10639	. . . . . . . 8 $/\$ $/\$
14		pntpbnd.r	. . . . . . . . . 10 ψ
15	14	pntrf 21210	. . . . . . . . 9
16	15	ffvelrni 5828	. . . . . . . 8
17	13, 16	syl 16	. . . . . . 7 $/\$ $/\$
18	17	recnd 9070	. . . . . 6 $/\$ $/\$
19	18	abscld 12193	. . . . 5 $/\$ $/\$
20	13	relogcld 20471	. . . . 5 $/\$ $/\$
21	19, 20	remulcld 9072	. . . 4 $/\$ $/\$
22		2re 10025	. . . . . . 7
23	22	a1i 11	. . . . . 6 $/\$ $/\$
24	5, 11	rplogcld 20477	. . . . . 6 $/\$ $/\$
25	23, 24	rerpdivcld 10631	. . . . 5 $/\$ $/\$
26		fzfid 11267	. . . . . 6 $/\$ $/\$
27	13	adantr 452	. . . . . . . . . . 11 $/\$ $/\$ $/\$
28		elfznn 11036	. . . . . . . . . . . . 13
29	28	adantl 453	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
30	29	nnrpd 10603	. . . . . . . . . . 11 $/\$ $/\$ $/\$
31	27, 30	rpdivcld 10621	. . . . . . . . . 10 $/\$ $/\$ $/\$
32	15	ffvelrni 5828	. . . . . . . . . 10
33	31, 32	syl 16	. . . . . . . . 9 $/\$ $/\$ $/\$
34	33	recnd 9070	. . . . . . . 8 $/\$ $/\$ $/\$
35	34	abscld 12193	. . . . . . 7 $/\$ $/\$ $/\$
36	30	relogcld 20471	. . . . . . 7 $/\$ $/\$ $/\$
37	35, 36	remulcld 9072	. . . . . 6 $/\$ $/\$ $/\$
38	26, 37	fsumrecl 12483	. . . . 5 $/\$ $/\$
39	25, 38	remulcld 9072	. . . 4 $/\$ $/\$
40	21, 39	resubcld 9421	. . 3 $/\$ $/\$
41	40, 13	rerpdivcld 10631	. 2 $/\$ $/\$
42	14	pntrmax 21211	. . 3
43		eqid 2404	. . . . 5 Λ ψ Λ ψ
44		eqid 2404	. . . . 5
45		simprl 733	. . . . 5 $/\$ $/\$ $/\$
46		simprr 734	. . . . 5 $/\$ $/\$ $/\$
47		simpll 731	. . . . 5 $/\$ $/\$ $/\$
48		simplr 732	. . . . 5 $/\$ $/\$ $/\$
49	43, 14, 44, 45, 46, 47, 48	pntrlog2bndlem6 21230	. . . 4 $/\$ $/\$ $/\$
50	49	rexlimdvaa 2791	. . 3 $/\$
51	42, 50	mpi 17	. 2 $/\$
52		simprl 733	. . . . 5 $/\$ $/\$ $/\$
53		chpcl 20860	. . . . 5 ψ
54	52, 53	syl 16	. . . 4 $/\$ $/\$ $/\$ ψ
55	54, 52	readdcld 9071	. . 3 $/\$ $/\$ $/\$ ψ
56	6	a1i 11	. . . . 5 $/\$ $/\$ $/\$
57		simprr 734	. . . . 5 $/\$ $/\$ $/\$
58	52, 56, 57	rpgecld 10639	. . . 4 $/\$ $/\$ $/\$
59	58	relogcld 20471	. . 3 $/\$ $/\$ $/\$
60	55, 59	remulcld 9072	. 2 $/\$ $/\$ $/\$ ψ
61	41	adantr 452	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
62	54	ad2ant2r 728	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
63		simprll 739	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
64	62, 63	readdcld 9071	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
65	58	ad2ant2r 728	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
66	65	relogcld 20471	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
67	64, 66	remulcld 9072	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
68	13	adantr 452	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
69	67, 68	rerpdivcld 10631	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
70	17	adantr 452	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
71	70	recnd 9070	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
72	71	abscld 12193	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
73	68	relogcld 20471	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
74	72, 73	remulcld 9072	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
75	25	adantr 452	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	38	adantr 452	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	75, 76	remulcld 9072	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
78	74, 77	resubcld 9421	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	22	a1i 11	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
80	5	adantr 452	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
81	11	adantr 452	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	rplogcld 20477	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
83		2rp 10573	. . . . . . . . . 10
84	83	a1i 11	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
85	84	rpge0d 10608	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
86	79, 82, 85	divge0d 10640	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
87		fzfid 11267	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
88	37	adantlr 696	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
89	34	adantlr 696	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
90	89	abscld 12193	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
91	30	adantlr 696	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
92	91	relogcld 20471	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
93	89	absge0d 12201	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
94	91	rpred 10604	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
95	28	adantl 453	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
96	95	nnge1d 9998	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
97	94, 96	logge0d 20478	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
98	90, 92, 93, 97	mulge0d 9559	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
99	87, 88, 98	fsumge0 12529	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
100	75, 76, 86, 99	mulge0d 9559	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
101	74, 77	subge02d 9574	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	100, 101	mpbid 202	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	71	absge0d 12201	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
104	82	rpge0d 10608	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
105		chpcl 20860	. . . . . . . . 9 ψ
106	80, 105	syl 16	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
107	106, 80	readdcld 9071	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
108	14	pntrval 21209	. . . . . . . . . 10 ψ
109	68, 108	syl 16	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
110	109	fveq2d 5691	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
111	106	recnd 9070	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
112	80	recnd 9070	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
113	111, 112	abs2dif2d 12215	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
114		chpge0 20862	. . . . . . . . . . . 12 ψ
115	80, 114	syl 16	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
116	106, 115	absidd 12180	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
117	68	rpge0d 10608	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
118	80, 117	absidd 12180	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
119	116, 118	oveq12d 6058	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
120	113, 119	breqtrd 4196	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
121	110, 120	eqbrtrd 4192	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
122		simprr 734	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
123	80, 63, 122	ltled 9177	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
124		chpwordi 20893	. . . . . . . . 9 $/\$ $/\$ ψ ψ
125	80, 63, 123, 124	syl3anc 1184	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
126	106, 80, 62, 63, 125, 123	le2addd 9600	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
127	72, 107, 64, 121, 126	letrd 9183	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
128	68, 65	logled 20475	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
129	123, 128	mpbid 202	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
130	72, 64, 73, 66, 103, 104, 127, 129	lemul12ad 9909	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
131	78, 74, 67, 102, 130	letrd 9183	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
132	78, 67, 68, 131	lediv1dd 10658	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
133	6	a1i 11	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
134		chpge0 20862	. . . . . . . 8 ψ
135	63, 134	syl 16	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
136	65	rpge0d 10608	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
137	62, 63, 135, 136	addge0d 9558	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
138		simprlr 740	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
139	63, 138	logge0d 20478	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
140	64, 66, 137, 139	mulge0d 9559	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
141	12	adantr 452	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
142	133, 68, 67, 140, 141	lediv2ad 10626	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
143	62	recnd 9070	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
144	63	recnd 9070	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
145	143, 144	addcld 9063	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
146	66	recnd 9070	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
147	145, 146	mulcld 9064	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
148	147	div1d 9738	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
149	142, 148	breqtrd 4196	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ ψ ψ
150	61, 69, 67, 132, 149	letrd 9183	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ ψ
151	2, 4, 41, 51, 60, 150	lo1bddrp 12274	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 359

wceq 1649

wcel 1721

wral 2666

wrex 2667

wss 3280

cif 3699 class class class wbr 4172

cmpt 4226

cfv 5413 (class class class)co 6040

cc 8944

cr 8945

cc0 8946

c1 8947

caddc 8949

cmul 8951

cpnf 9073

clt 9076

cle 9077

cmin 9247

cdiv 9633

cn 9956

c2 10005

crp 10568

cioo 10872

cfz 10999

cfl 11156

cabs 11994

clo1 12236

csu 12434

clog 20405 Λcvma 20827 ψcchp 20828

This theorem is referenced by: pntlemp 21257

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 ax-inf2 7552 ax-cnex 9002 ax-resscn 9003 ax-1cn 9004 ax-icn 9005 ax-addcl 9006 ax-addrcl 9007 ax-mulcl 9008 ax-mulrcl 9009 ax-mulcom 9010 ax-addass 9011 ax-mulass 9012 ax-distr 9013 ax-i2m1 9014 ax-1ne0 9015 ax-1rid 9016 ax-rnegex 9017 ax-rrecex 9018 ax-cnre 9019 ax-pre-lttri 9020 ax-pre-lttrn 9021 ax-pre-ltadd 9022 ax-pre-mulgt0 9023 ax-pre-sup 9024 ax-addf 9025 ax-mulf 9026

This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-pss 3296 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-tp 3782 df-op 3783 df-uni 3976 df-int 4011 df-iun 4055 df-iin 4056 df-disj 4143 df-br 4173 df-opab 4227 df-mpt 4228 df-tr 4263 df-eprel 4454 df-id 4458 df-po 4463 df-so 4464 df-fr 4501 df-se 4502 df-we 4503 df-ord 4544 df-on 4545 df-lim 4546 df-suc 4547 df-om 4805 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-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-isom 5422 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-of 6264 df-1st 6308 df-2nd 6309 df-riota 6508 df-recs 6592 df-rdg 6627 df-1o 6683 df-2o 6684 df-oadd 6687 df-er 6864 df-map 6979 df-pm 6980 df-ixp 7023 df-en 7069 df-dom 7070 df-sdom 7071 df-fin 7072 df-fi 7374 df-sup 7404 df-oi 7435 df-card 7782 df-cda 8004 df-pnf 9078 df-mnf 9079 df-xr 9080 df-ltxr 9081 df-le 9082 df-sub 9249 df-neg 9250 df-div 9634 df-nn 9957 df-2 10014 df-3 10015 df-4 10016 df-5 10017 df-6 10018 df-7 10019 df-8 10020 df-9 10021 df-10 10022 df-n0 10178 df-z 10239 df-dec 10339 df-uz 10445 df-q 10531 df-rp 10569 df-xneg 10666 df-xadd 10667 df-xmul 10668 df-ioo 10876 df-ioc 10877 df-ico 10878 df-icc 10879 df-fz 11000 df-fzo 11091 df-fl 11157 df-mod 11206 df-seq 11279 df-exp 11338 df-fac 11522 df-bc 11549 df-hash 11574 df-shft 11837 df-cj 11859 df-re 11860 df-im 11861 df-sqr 11995 df-abs 11996 df-limsup 12220 df-clim 12237 df-rlim 12238 df-o1 12239 df-lo1 12240 df-sum 12435 df-ef 12625 df-e 12626 df-sin 12627 df-cos 12628 df-pi 12630 df-dvds 12808 df-gcd 12962 df-prm 13035 df-pc 13166 df-struct 13426 df-ndx 13427 df-slot 13428 df-base 13429 df-sets 13430 df-ress 13431 df-plusg 13497 df-mulr 13498 df-starv 13499 df-sca 13500 df-vsca 13501 df-tset 13503 df-ple 13504 df-ds 13506 df-unif 13507 df-hom 13508 df-cco 13509 df-rest 13605 df-topn 13606 df-topgen 13622 df-pt 13623 df-prds 13626 df-xrs 13681 df-0g 13682 df-gsum 13683 df-qtop 13688 df-imas 13689 df-xps 13691 df-mre 13766 df-mrc 13767 df-acs 13769 df-mnd 14645 df-submnd 14694 df-mulg 14770 df-cntz 15071 df-cmn 15369 df-psmet 16649 df-xmet 16650 df-met 16651 df-bl 16652 df-mopn 16653 df-fbas 16654 df-fg 16655 df-cnfld 16659 df-top 16918 df-bases 16920 df-topon 16921 df-topsp 16922 df-cld 17038 df-ntr 17039 df-cls 17040 df-nei 17117 df-lp 17155 df-perf 17156 df-cn 17245 df-cnp 17246 df-haus 17333 df-cmp 17404 df-tx 17547 df-hmeo 17740 df-fil 17831 df-fm 17923 df-flim 17924 df-flf 17925 df-xms 18303 df-ms 18304 df-tms 18305 df-cncf 18861 df-limc 19706 df-dv 19707 df-log 20407 df-cxp 20408 df-em 20784 df-cht 20832 df-vma 20833 df-chp 20834 df-ppi 20835 df-mu 20836

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