logfaclbnd - Metamath Proof Explorer

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Theorem logfaclbnd 24947

Description: A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

**Assertion**
Ref	Expression
logfaclbnd

Proof of Theorem logfaclbnd Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpcn 11841	. . . . 5
2	1	times2d 11276	. . . 4
3	2	oveq2d 6666	. . 3
4		relogcl 24322	. . . . 5
5	4	recnd 10068	. . . 4
6		2cnd 11093	. . . 4
7	1, 5, 6	subdid 10486	. . 3
8		rpre 11839	. . . . . 6
9	8, 4	remulcld 10070	. . . . 5
10	9	recnd 10068	. . . 4
11	10, 1, 1	subsub4d 10423	. . 3
12	3, 7, 11	3eqtr4d 2666	. 2
13	9, 8	resubcld 10458	. . . 4
14		fzfid 12772	. . . . 5
15		fzfid 12772	. . . . . 6 $/\$
16		elfznn 12370	. . . . . . . 8
17	16	adantl 482	. . . . . . 7 $/\$ $/\$
18	17	nnrecred 11066	. . . . . 6 $/\$ $/\$
19	15, 18	fsumrecl 14465	. . . . 5 $/\$
20	14, 19	fsumrecl 14465	. . . 4
21		rprege0 11847	. . . . . . . . 9 $/\$
22		flge0nn0 12621	. . . . . . . . 9 $/\$
23	21, 22	syl 17	. . . . . . . 8
24		faccl 13070	. . . . . . . 8
25	23, 24	syl 17	. . . . . . 7
26	25	nnrpd 11870	. . . . . 6
27	26	relogcld 24369	. . . . 5
28	27, 8	readdcld 10069	. . . 4
29		elfznn 12370	. . . . . . . . . 10
30	29	adantl 482	. . . . . . . . 9 $/\$
31	30	nnrecred 11066	. . . . . . . 8 $/\$
32	14, 31	fsumrecl 14465	. . . . . . 7
33	8, 32	remulcld 10070	. . . . . 6
34		reflcl 12597	. . . . . . 7
35	8, 34	syl 17	. . . . . 6
36	33, 35	resubcld 10458	. . . . 5
37		harmoniclbnd 24735	. . . . . . 7
38		rpregt0 11846	. . . . . . . 8 $/\$
39		lemul2 10876	. . . . . . . 8 $/\$ $/\$ $/\$
40	4, 32, 38, 39	syl3anc 1326	. . . . . . 7
41	37, 40	mpbid 222	. . . . . 6
42		flle 12600	. . . . . . 7
43	8, 42	syl 17	. . . . . 6
44	9, 35, 33, 8, 41, 43	le2subd 10647	. . . . 5
45	29	nnrecred 11066	. . . . . . . . 9
46		remulcl 10021	. . . . . . . . 9 $/\$
47	8, 45, 46	syl2an 494	. . . . . . . 8 $/\$
48		peano2rem 10348	. . . . . . . 8
49	47, 48	syl 17	. . . . . . 7 $/\$
50		fzfid 12772	. . . . . . . 8 $/\$
51	31	adantr 481	. . . . . . . 8 $/\$ $/\$
52	50, 51	fsumrecl 14465	. . . . . . 7 $/\$
53	8	adantr 481	. . . . . . . . . 10 $/\$
54	53, 34	syl 17	. . . . . . . . . . 11 $/\$
55		peano2re 10209	. . . . . . . . . . 11
56	54, 55	syl 17	. . . . . . . . . 10 $/\$
57	30	nnred 11035	. . . . . . . . . 10 $/\$
58		fllep1 12602	. . . . . . . . . . . 12
59	8, 58	syl 17	. . . . . . . . . . 11
60	59	adantr 481	. . . . . . . . . 10 $/\$
61	53, 56, 57, 60	lesub1dd 10643	. . . . . . . . 9 $/\$
62	53, 57	resubcld 10458	. . . . . . . . . 10 $/\$
63	56, 57	resubcld 10458	. . . . . . . . . 10 $/\$
64	30	nnrpd 11870	. . . . . . . . . . 11 $/\$
65	64	rpreccld 11882	. . . . . . . . . 10 $/\$
66	62, 63, 65	lemul1d 11915	. . . . . . . . 9 $/\$
67	61, 66	mpbid 222	. . . . . . . 8 $/\$
68	1	adantr 481	. . . . . . . . . 10 $/\$
69	30	nncnd 11036	. . . . . . . . . 10 $/\$
70	31	recnd 10068	. . . . . . . . . 10 $/\$
71	68, 69, 70	subdird 10487	. . . . . . . . 9 $/\$
72	30	nnne0d 11065	. . . . . . . . . . 11 $/\$
73	69, 72	recidd 10796	. . . . . . . . . 10 $/\$
74	73	oveq2d 6666	. . . . . . . . 9 $/\$
75	71, 74	eqtr2d 2657	. . . . . . . 8 $/\$
76		fsumconst 14522	. . . . . . . . . 10 $/\$
77	50, 70, 76	syl2anc 693	. . . . . . . . 9 $/\$
78		elfzuz3 12339	. . . . . . . . . . . . 13
79	78	adantl 482	. . . . . . . . . . . 12 $/\$
80		hashfz 13214	. . . . . . . . . . . 12
81	79, 80	syl 17	. . . . . . . . . . 11 $/\$
82	35	recnd 10068	. . . . . . . . . . . . 13
83	82	adantr 481	. . . . . . . . . . . 12 $/\$
84		1cnd 10056	. . . . . . . . . . . 12 $/\$
85	83, 84, 69	addsubd 10413	. . . . . . . . . . 11 $/\$
86	81, 85	eqtr4d 2659	. . . . . . . . . 10 $/\$
87	86	oveq1d 6665	. . . . . . . . 9 $/\$
88	77, 87	eqtrd 2656	. . . . . . . 8 $/\$
89	67, 75, 88	3brtr4d 4685	. . . . . . 7 $/\$
90	14, 49, 52, 89	fsumle 14531	. . . . . 6
91	14, 1, 70	fsummulc2 14516	. . . . . . . 8
92		ax-1cn 9994	. . . . . . . . . 10
93		fsumconst 14522	. . . . . . . . . 10 $/\$
94	14, 92, 93	sylancl 694	. . . . . . . . 9
95		hashfz1 13134	. . . . . . . . . . 11
96	23, 95	syl 17	. . . . . . . . . 10
97	96	oveq1d 6665	. . . . . . . . 9
98	82	mulid1d 10057	. . . . . . . . 9
99	94, 97, 98	3eqtrrd 2661	. . . . . . . 8
100	91, 99	oveq12d 6668	. . . . . . 7
101	47	recnd 10068	. . . . . . . 8 $/\$
102	14, 101, 84	fsumsub 14520	. . . . . . 7
103	100, 102	eqtr4d 2659	. . . . . 6
104		eqid 2622	. . . . . . . . . . . . . 14
105	104	uztrn2 11705	. . . . . . . . . . . . 13 $/\$
106	105	adantl 482	. . . . . . . . . . . 12 $/\$ $/\$
107	106	biantrurd 529	. . . . . . . . . . 11 $/\$ $/\$ $/\$
108		uzss 11708	. . . . . . . . . . . . . 14
109	108	ad2antll 765	. . . . . . . . . . . . 13 $/\$ $/\$
110	109	sseld 3602	. . . . . . . . . . . 12 $/\$ $/\$
111	110	pm4.71rd 667	. . . . . . . . . . 11 $/\$ $/\$ $/\$
112	107, 111	bitr3d 270	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
113	112	pm5.32da 673	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
114		ancom 466	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
115		an4 865	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
116	113, 114, 115	3bitr4g 303	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
117		elfzuzb 12336	. . . . . . . . 9 $/\$
118		elfzuzb 12336	. . . . . . . . 9 $/\$
119	117, 118	anbi12i 733	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
120		elfzuzb 12336	. . . . . . . . 9 $/\$
121		elfzuzb 12336	. . . . . . . . 9 $/\$
122	120, 121	anbi12i 733	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
123	116, 119, 122	3bitr4g 303	. . . . . . 7 $/\$ $/\$
124	18	recnd 10068	. . . . . . . 8 $/\$ $/\$
125	124	anasss 679	. . . . . . 7 $/\$ $/\$
126	14, 14, 15, 123, 125	fsumcom2 14505	. . . . . 6
127	90, 103, 126	3brtr4d 4685	. . . . 5
128	13, 36, 20, 44, 127	letrd 10194	. . . 4
129	27, 35	readdcld 10069	. . . . 5
130		elfznn 12370	. . . . . . . . . . 11
131	130	adantl 482	. . . . . . . . . 10 $/\$
132	131	nnrpd 11870	. . . . . . . . 9 $/\$
133	132	relogcld 24369	. . . . . . . 8 $/\$
134		peano2re 10209	. . . . . . . 8
135	133, 134	syl 17	. . . . . . 7 $/\$
136		nnz 11399	. . . . . . . . . . . 12
137		flid 12609	. . . . . . . . . . . 12
138	136, 137	syl 17	. . . . . . . . . . 11
139	138	oveq2d 6666	. . . . . . . . . 10
140	139	sumeq1d 14431	. . . . . . . . 9
141		nnre 11027	. . . . . . . . . 10
142		nnge1 11046	. . . . . . . . . 10
143		harmonicubnd 24736	. . . . . . . . . 10 $/\$
144	141, 142, 143	syl2anc 693	. . . . . . . . 9
145	140, 144	eqbrtrrd 4677	. . . . . . . 8
146	131, 145	syl 17	. . . . . . 7 $/\$
147	14, 19, 135, 146	fsumle 14531	. . . . . 6
148	133	recnd 10068	. . . . . . . 8 $/\$
149		1cnd 10056	. . . . . . . 8 $/\$
150	14, 148, 149	fsumadd 14470	. . . . . . 7
151		logfac 24347	. . . . . . . . 9
152	23, 151	syl 17	. . . . . . . 8
153		fsumconst 14522	. . . . . . . . . 10 $/\$
154	14, 92, 153	sylancl 694	. . . . . . . . 9
155	154, 97, 98	3eqtrrd 2661	. . . . . . . 8
156	152, 155	oveq12d 6668	. . . . . . 7
157	150, 156	eqtr4d 2659	. . . . . 6
158	147, 157	breqtrd 4679	. . . . 5
159	35, 8, 27, 43	leadd2dd 10642	. . . . 5
160	20, 129, 28, 158, 159	letrd 10194	. . . 4
161	13, 20, 28, 128, 160	letrd 10194	. . 3
162	13, 8, 27	lesubaddd 10624	. . 3
163	161, 162	mpbird 247	. 2
164	12, 163	eqbrtrd 4675	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wss 3574 class class class wbr 4653

cfv 5888 (class class class)co 6650

cfn 7955

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cdiv 10684

cn 11020

c2 11070

cn0 11292

cz 11377

cuz 11687

crp 11832

cfz 12326

cfl 12591

cfa 13060

chash 13117

csu 14416

clog 24301

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-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

This theorem is referenced by: (None)

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