chebbnd1lem1 - Metamath Proof Explorer

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Theorem chebbnd1lem1 25158

Description: Lemma for chebbnd1 25161: show a lower bound on π

at even integers using similar techniques to those used to prove bpos 25018. (Note that the expression

is actually equal to

, but proving that is not necessary for the proof, and it's too much work.) The key to the proof is bposlem1 25009, which shows that each term in the expansion

is at most

, so that the sum really only has nonzero elements up to

, and since each term is at most

, after taking logs we get the inequality π

, and bclbnd 25005 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)

**Hypothesis**
Ref	Expression
chebbnd1lem1.1

**Assertion**
Ref	Expression
chebbnd1lem1	π

Proof of Theorem chebbnd1lem1 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4nn 11187	. . . . . 6
2		eluznn 11758	. . . . . . . 8 $/\$
3	1, 2	mpan 706	. . . . . . 7
4	3	nnnn0d 11351	. . . . . 6
5		nnexpcl 12873	. . . . . 6 $/\$
6	1, 4, 5	sylancr 695	. . . . 5
7	6	nnrpd 11870	. . . 4
8	3	nnrpd 11870	. . . 4
9	7, 8	rpdivcld 11889	. . 3
10	9	relogcld 24369	. 2
11		fzctr 12451	. . . . . 6
12	4, 11	syl 17	. . . . 5
13		bccl2 13110	. . . . 5
14	12, 13	syl 17	. . . 4
15	14	nnrpd 11870	. . 3
16	15	relogcld 24369	. 2
17		2z 11409	. . . . . . 7
18		eluzelz 11697	. . . . . . 7
19		zmulcl 11426	. . . . . . 7 $/\$
20	17, 18, 19	sylancr 695	. . . . . 6
21	20	zred 11482	. . . . 5
22		ppicl 24857	. . . . 5 π
23	21, 22	syl 17	. . . 4 π
24	23	nn0red 11352	. . 3 π
25		2nn 11185	. . . . . 6
26		nnmulcl 11043	. . . . . 6 $/\$
27	25, 3, 26	sylancr 695	. . . . 5
28	27	nnrpd 11870	. . . 4
29	28	relogcld 24369	. . 3
30	24, 29	remulcld 10070	. 2 π
31		bclbnd 25005	. . 3
32		logltb 24346	. . . 4 $/\$
33	9, 15, 32	syl2anc 693	. . 3
34	31, 33	mpbid 222	. 2
35		chebbnd1lem1.1	. . . . . . . 8
36	27, 14	ifcld 4131	. . . . . . . 8
37	35, 36	syl5eqel 2705	. . . . . . 7
38	37	nnred 11035	. . . . . 6
39		ppicl 24857	. . . . . 6 π
40	38, 39	syl 17	. . . . 5 π
41	40	nn0red 11352	. . . 4 π
42	41, 29	remulcld 10070	. . 3 π
43		fzfid 12772	. . . . . 6
44		inss1 3833	. . . . . 6
45		ssfi 8180	. . . . . 6 $/\$
46	43, 44, 45	sylancl 694	. . . . 5
47	37	nnzd 11481	. . . . . . . . . 10
48	14	nnzd 11481	. . . . . . . . . 10
49	14	nnred 11035	. . . . . . . . . . . 12
50		min2 12021	. . . . . . . . . . . 12 $/\$
51	21, 49, 50	syl2anc 693	. . . . . . . . . . 11
52	35, 51	syl5eqbr 4688	. . . . . . . . . 10
53		eluz2 11693	. . . . . . . . . 10 $/\$ $/\$
54	47, 48, 52, 53	syl3anbrc 1246	. . . . . . . . 9
55		fzss2 12381	. . . . . . . . 9
56	54, 55	syl 17	. . . . . . . 8
57		ssrin 3838	. . . . . . . 8
58	56, 57	syl 17	. . . . . . 7
59	58	sselda 3603	. . . . . 6 $/\$
60		inss1 3833	. . . . . . . . . . 11
61		simpr 477	. . . . . . . . . . 11 $/\$
62	60, 61	sseldi 3601	. . . . . . . . . 10 $/\$
63		elfznn 12370	. . . . . . . . . 10
64	62, 63	syl 17	. . . . . . . . 9 $/\$
65		inss2 3834	. . . . . . . . . . 11
66	65, 61	sseldi 3601	. . . . . . . . . 10 $/\$
67	14	adantr 481	. . . . . . . . . 10 $/\$
68	66, 67	pccld 15555	. . . . . . . . 9 $/\$
69	64, 68	nnexpcld 13030	. . . . . . . 8 $/\$
70	69	nnrpd 11870	. . . . . . 7 $/\$
71	70	relogcld 24369	. . . . . 6 $/\$
72	59, 71	syldan 487	. . . . 5 $/\$
73	29	adantr 481	. . . . 5 $/\$
74		elin 3796	. . . . . . . . 9 $/\$
75	74	simprbi 480	. . . . . . . 8
76		bposlem1 25009	. . . . . . . 8 $/\$
77	3, 75, 76	syl2an 494	. . . . . . 7 $/\$
78	59, 70	syldan 487	. . . . . . . 8 $/\$
79	78	reeflogd 24370	. . . . . . 7 $/\$
80	28	adantr 481	. . . . . . . 8 $/\$
81	80	reeflogd 24370	. . . . . . 7 $/\$
82	77, 79, 81	3brtr4d 4685	. . . . . 6 $/\$
83		efle 14848	. . . . . . 7 $/\$
84	72, 73, 83	syl2anc 693	. . . . . 6 $/\$
85	82, 84	mpbird 247	. . . . 5 $/\$
86	46, 72, 73, 85	fsumle 14531	. . . 4
87	71	recnd 10068	. . . . . . 7 $/\$
88	59, 87	syldan 487	. . . . . 6 $/\$
89		eldifn 3733	. . . . . . . . . . . . 13 $\$
90	89	adantl 482	. . . . . . . . . . . 12 $/\$ $\$
91		simpr 477	. . . . . . . . . . . . . . . . . . 19 $/\$ $\$ $\$
92	91	eldifad 3586	. . . . . . . . . . . . . . . . . 18 $/\$ $\$
93	60, 92	sseldi 3601	. . . . . . . . . . . . . . . . 17 $/\$ $\$
94	93, 63	syl 17	. . . . . . . . . . . . . . . 16 $/\$ $\$
95	94	adantrr 753	. . . . . . . . . . . . . . 15 $/\$ $\$ $/\$
96	95	nnred 11035	. . . . . . . . . . . . . . . . . 18 $/\$ $\$ $/\$
97	92, 69	syldan 487	. . . . . . . . . . . . . . . . . . . 20 $/\$ $\$
98	97	nnred 11035	. . . . . . . . . . . . . . . . . . 19 $/\$ $\$
99	98	adantrr 753	. . . . . . . . . . . . . . . . . 18 $/\$ $\$ $/\$
100	21	adantr 481	. . . . . . . . . . . . . . . . . 18 $/\$ $\$ $/\$
101	95	nncnd 11036	. . . . . . . . . . . . . . . . . . . 20 $/\$ $\$ $/\$
102	101	exp1d 13003	. . . . . . . . . . . . . . . . . . 19 $/\$ $\$ $/\$
103	95	nnge1d 11063	. . . . . . . . . . . . . . . . . . . 20 $/\$ $\$ $/\$
104		simprr 796	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $\$ $/\$
105		nnuz 11723	. . . . . . . . . . . . . . . . . . . . 21
106	104, 105	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . 20 $/\$ $\$ $/\$
107	96, 103, 106	leexp2ad 13041	. . . . . . . . . . . . . . . . . . 19 $/\$ $\$ $/\$
108	102, 107	eqbrtrrd 4677	. . . . . . . . . . . . . . . . . 18 $/\$ $\$ $/\$
109	3	adantr 481	. . . . . . . . . . . . . . . . . . 19 $/\$ $\$ $/\$
110	65, 92	sseldi 3601	. . . . . . . . . . . . . . . . . . . 20 $/\$ $\$
111	110	adantrr 753	. . . . . . . . . . . . . . . . . . 19 $/\$ $\$ $/\$
112	109, 111, 76	syl2anc 693	. . . . . . . . . . . . . . . . . 18 $/\$ $\$ $/\$
113	96, 99, 100, 108, 112	letrd 10194	. . . . . . . . . . . . . . . . 17 $/\$ $\$ $/\$
114		elfzle2 12345	. . . . . . . . . . . . . . . . . . 19
115	93, 114	syl 17	. . . . . . . . . . . . . . . . . 18 $/\$ $\$
116	115	adantrr 753	. . . . . . . . . . . . . . . . 17 $/\$ $\$ $/\$
117	49	adantr 481	. . . . . . . . . . . . . . . . . 18 $/\$ $\$ $/\$
118		lemin 12023	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
119	96, 100, 117, 118	syl3anc 1326	. . . . . . . . . . . . . . . . 17 $/\$ $\$ $/\$ $/\$
120	113, 116, 119	mpbir2and 957	. . . . . . . . . . . . . . . 16 $/\$ $\$ $/\$
121	120, 35	syl6breqr 4695	. . . . . . . . . . . . . . 15 $/\$ $\$ $/\$
122	37	adantr 481	. . . . . . . . . . . . . . . . 17 $/\$ $\$ $/\$
123	122	nnzd 11481	. . . . . . . . . . . . . . . 16 $/\$ $\$ $/\$
124		fznn 12408	. . . . . . . . . . . . . . . 16 $/\$
125	123, 124	syl 17	. . . . . . . . . . . . . . 15 $/\$ $\$ $/\$ $/\$
126	95, 121, 125	mpbir2and 957	. . . . . . . . . . . . . 14 $/\$ $\$ $/\$
127	126, 111	elind 3798	. . . . . . . . . . . . 13 $/\$ $\$ $/\$
128	127	expr 643	. . . . . . . . . . . 12 $/\$ $\$
129	90, 128	mtod 189	. . . . . . . . . . 11 $/\$ $\$
130	92, 68	syldan 487	. . . . . . . . . . . . 13 $/\$ $\$
131		elnn0 11294	. . . . . . . . . . . . 13 $\/$
132	130, 131	sylib 208	. . . . . . . . . . . 12 $/\$ $\$ $\/$
133	132	ord 392	. . . . . . . . . . 11 $/\$ $\$
134	129, 133	mpd 15	. . . . . . . . . 10 $/\$ $\$
135	134	oveq2d 6666	. . . . . . . . 9 $/\$ $\$
136	94	nncnd 11036	. . . . . . . . . 10 $/\$ $\$
137	136	exp0d 13002	. . . . . . . . 9 $/\$ $\$
138	135, 137	eqtrd 2656	. . . . . . . 8 $/\$ $\$
139	138	fveq2d 6195	. . . . . . 7 $/\$ $\$
140		log1 24332	. . . . . . 7
141	139, 140	syl6eq 2672	. . . . . 6 $/\$ $\$
142		fzfid 12772	. . . . . . 7
143		ssfi 8180	. . . . . . 7 $/\$
144	142, 60, 143	sylancl 694	. . . . . 6
145	58, 88, 141, 144	fsumss 14456	. . . . 5
146	64	nnrpd 11870	. . . . . . 7 $/\$
147	68	nn0zd 11480	. . . . . . 7 $/\$
148		relogexp 24342	. . . . . . 7 $/\$
149	146, 147, 148	syl2anc 693	. . . . . 6 $/\$
150	149	sumeq2dv 14433	. . . . 5
151		pclogsum 24940	. . . . . 6
152	14, 151	syl 17	. . . . 5
153	145, 150, 152	3eqtrd 2660	. . . 4
154	29	recnd 10068	. . . . . 6
155		fsumconst 14522	. . . . . 6 $/\$
156	46, 154, 155	syl2anc 693	. . . . 5
157		2eluzge1 11734	. . . . . . 7
158		ppival2g 24855	. . . . . . 7 $/\$ π
159	47, 157, 158	sylancl 694	. . . . . 6 π
160	159	oveq1d 6665	. . . . 5 π
161	156, 160	eqtr4d 2659	. . . 4 π
162	86, 153, 161	3brtr3d 4684	. . 3 π
163		min1 12020	. . . . . . 7 $/\$
164	21, 49, 163	syl2anc 693	. . . . . 6
165	35, 164	syl5eqbr 4688	. . . . 5
166		ppiwordi 24888	. . . . 5 $/\$ $/\$ π π
167	38, 21, 165, 166	syl3anc 1326	. . . 4 π π
168		1red 10055	. . . . . . 7
169		2re 11090	. . . . . . . 8
170	169	a1i 11	. . . . . . 7
171		1lt2 11194	. . . . . . . 8
172	171	a1i 11	. . . . . . 7
173		2t1e2 11176	. . . . . . . 8
174	3	nnge1d 11063	. . . . . . . . 9
175		eluzelre 11698	. . . . . . . . . 10
176		2pos 11112	. . . . . . . . . . . 12
177	169, 176	pm3.2i 471	. . . . . . . . . . 11 $/\$
178	177	a1i 11	. . . . . . . . . 10 $/\$
179		lemul2 10876	. . . . . . . . . 10 $/\$ $/\$ $/\$
180	168, 175, 178, 179	syl3anc 1326	. . . . . . . . 9
181	174, 180	mpbid 222	. . . . . . . 8
182	173, 181	syl5eqbrr 4689	. . . . . . 7
183	168, 170, 21, 172, 182	ltletrd 10197	. . . . . 6
184	21, 183	rplogcld 24375	. . . . 5
185	41, 24, 184	lemul1d 11915	. . . 4 π π π π
186	167, 185	mpbid 222	. . 3 π π
187	16, 42, 30, 162, 186	letrd 10194	. 2 π
188	10, 16, 30, 34, 187	ltletrd 10197	1 π

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990 $\$ cdif 3571

cin 3573

wss 3574

cif 4086 class class class wbr 4653

cfv 5888 (class class class)co 6650

cfn 7955

cc 9934

cr 9935

cc0 9936

c1 9937

cmul 9941

clt 10074

cle 10075

cdiv 10684

cn 11020

c2 11070

c4 11072

cn0 11292

cz 11377

cuz 11687

crp 11832

cfz 12326

cexp 12860

cbc 13089

chash 13117

csu 14416

ce 14792

cprime 15385

cpc 15541

clog 24301 πcppi 24820

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-xnn0 11364 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-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-ppi 24826

This theorem is referenced by: chebbnd1lem3 25160

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