Theorem logdivlti 24366

Description: The log x / x function is strictly decreasing on the reals greater than _e. (Contributed by Mario Carneiro, 14-Mar-2014.)

Assertion

Ref	Expression
logdivlti	|- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` B ) / B ) < ( ( log ` A ) / A ) )

Proof of Theorem logdivlti

Step	Hyp	Ref	Expression
1		simpl2 1065	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> B e. RR )
2		simpl3 1066	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> _e <_ A )
3		simpr 477	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A < B )
4		simpl1 1064	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A e. RR )
5		ere 14819	. . . . . . . . . . . 12 |- _e e. RR
6		lelttr 10128	. . . . . . . . . . . 12 |- ( ( _e e. RR /\ A e. RR /\ B e. RR ) -> ( ( _e <_ A /\ A < B ) -> _e < B ) )
7	5, 6	mp3an1 1411	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( ( _e <_ A /\ A < B ) -> _e < B ) )
8	4, 1, 7	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( _e <_ A /\ A < B ) -> _e < B ) )
9	2, 3, 8	mp2and 715	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> _e < B )
10		epos 14935	. . . . . . . . . 10 |- 0 < _e
11		0re 10040	. . . . . . . . . . . 12 |- 0 e. RR
12		lttr 10114	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ _e e. RR /\ B e. RR ) -> ( ( 0 < _e /\ _e < B ) -> 0 < B ) )
13	11, 5, 12	mp3an12 1414	. . . . . . . . . . 11 |- ( B e. RR -> ( ( 0 < _e /\ _e < B ) -> 0 < B ) )
14	1, 13	syl 17	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( 0 < _e /\ _e < B ) -> 0 < B ) )
15	10, 14	mpani 712	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( _e < B -> 0 < B ) )
16	9, 15	mpd 15	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 0 < B )
17	1, 16	elrpd 11869	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> B e. RR+ )
18		ltletr 10129	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ _e e. RR /\ A e. RR ) -> ( ( 0 < _e /\ _e <_ A ) -> 0 < A ) )
19	11, 5, 18	mp3an12 1414	. . . . . . . . . . 11 |- ( A e. RR -> ( ( 0 < _e /\ _e <_ A ) -> 0 < A ) )
20	4, 19	syl 17	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( 0 < _e /\ _e <_ A ) -> 0 < A ) )
21	10, 20	mpani 712	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( _e <_ A -> 0 < A ) )
22	2, 21	mpd 15	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 0 < A )
23	4, 22	elrpd 11869	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A e. RR+ )
24	17, 23	rpdivcld 11889	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( B / A ) e. RR+ )
25		relogcl 24322	. . . . . 6 |- ( ( B / A ) e. RR+ -> ( log ` ( B / A ) ) e. RR )
26	24, 25	syl 17	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` ( B / A ) ) e. RR )
27	1, 23	rerpdivcld 11903	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( B / A ) e. RR )
28		1re 10039	. . . . . 6 |- 1 e. RR
29		resubcl 10345	. . . . . 6 |- ( ( ( B / A ) e. RR /\ 1 e. RR ) -> ( ( B / A ) - 1 ) e. RR )
30	27, 28, 29	sylancl 694	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) - 1 ) e. RR )
31		relogcl 24322	. . . . . . 7 |- ( A e. RR+ -> ( log ` A ) e. RR )
32	23, 31	syl 17	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` A ) e. RR )
33	30, 32	remulcld 10070	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. ( log ` A ) ) e. RR )
34		reeflog 24327	. . . . . . . . 9 |- ( ( B / A ) e. RR+ -> ( exp ` ( log ` ( B / A ) ) ) = ( B / A ) )
35	24, 34	syl 17	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` ( log ` ( B / A ) ) ) = ( B / A ) )
36		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
37	27	recnd 10068	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( B / A ) e. CC )
38		pncan3 10289	. . . . . . . . 9 |- ( ( 1 e. CC /\ ( B / A ) e. CC ) -> ( 1 + ( ( B / A ) - 1 ) ) = ( B / A ) )
39	36, 37, 38	sylancr 695	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 + ( ( B / A ) - 1 ) ) = ( B / A ) )
40	35, 39	eqtr4d 2659	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` ( log ` ( B / A ) ) ) = ( 1 + ( ( B / A ) - 1 ) ) )
41	4	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A e. CC )
42	41	mulid2d 10058	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 x. A ) = A )
43	42, 3	eqbrtrd 4675	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 x. A ) < B )
44		1red 10055	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 1 e. RR )
45		ltmuldiv 10896	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
46	44, 1, 4, 22, 45	syl112anc 1330	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
47	43, 46	mpbid 222	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 1 < ( B / A ) )
48		difrp 11868	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( B / A ) e. RR ) -> ( 1 < ( B / A ) <-> ( ( B / A ) - 1 ) e. RR+ ) )
49	28, 27, 48	sylancr 695	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 < ( B / A ) <-> ( ( B / A ) - 1 ) e. RR+ ) )
50	47, 49	mpbid 222	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) - 1 ) e. RR+ )
51		efgt1p 14845	. . . . . . . 8 |- ( ( ( B / A ) - 1 ) e. RR+ -> ( 1 + ( ( B / A ) - 1 ) ) < ( exp ` ( ( B / A ) - 1 ) ) )
52	50, 51	syl 17	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 + ( ( B / A ) - 1 ) ) < ( exp ` ( ( B / A ) - 1 ) ) )
53	40, 52	eqbrtrd 4675	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` ( log ` ( B / A ) ) ) < ( exp ` ( ( B / A ) - 1 ) ) )
54		eflt 14847	. . . . . . 7 |- ( ( ( log ` ( B / A ) ) e. RR /\ ( ( B / A ) - 1 ) e. RR ) -> ( ( log ` ( B / A ) ) < ( ( B / A ) - 1 ) <-> ( exp ` ( log ` ( B / A ) ) ) < ( exp ` ( ( B / A ) - 1 ) ) ) )
55	26, 30, 54	syl2anc 693	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` ( B / A ) ) < ( ( B / A ) - 1 ) <-> ( exp ` ( log ` ( B / A ) ) ) < ( exp ` ( ( B / A ) - 1 ) ) ) )
56	53, 55	mpbird 247	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` ( B / A ) ) < ( ( B / A ) - 1 ) )
57	30	recnd 10068	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) - 1 ) e. CC )
58	57	mulid1d 10057	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. 1 ) = ( ( B / A ) - 1 ) )
59		df-e 14799	. . . . . . . . 9 |- _e = ( exp ` 1 )
60		reeflog 24327	. . . . . . . . . . 11 |- ( A e. RR+ -> ( exp ` ( log ` A ) ) = A )
61	23, 60	syl 17	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` ( log ` A ) ) = A )
62	2, 61	breqtrrd 4681	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> _e <_ ( exp ` ( log ` A ) ) )
63	59, 62	syl5eqbrr 4689	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` 1 ) <_ ( exp ` ( log ` A ) ) )
64		efle 14848	. . . . . . . . 9 |- ( ( 1 e. RR /\ ( log ` A ) e. RR ) -> ( 1 <_ ( log ` A ) <-> ( exp ` 1 ) <_ ( exp ` ( log ` A ) ) ) )
65	28, 32, 64	sylancr 695	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 <_ ( log ` A ) <-> ( exp ` 1 ) <_ ( exp ` ( log ` A ) ) ) )
66	63, 65	mpbird 247	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 1 <_ ( log ` A ) )
67		posdif 10521	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( B / A ) e. RR ) -> ( 1 < ( B / A ) <-> 0 < ( ( B / A ) - 1 ) ) )
68	28, 27, 67	sylancr 695	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 < ( B / A ) <-> 0 < ( ( B / A ) - 1 ) ) )
69	47, 68	mpbid 222	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 0 < ( ( B / A ) - 1 ) )
70		lemul2 10876	. . . . . . . 8 |- ( ( 1 e. RR /\ ( log ` A ) e. RR /\ ( ( ( B / A ) - 1 ) e. RR /\ 0 < ( ( B / A ) - 1 ) ) ) -> ( 1 <_ ( log ` A ) <-> ( ( ( B / A ) - 1 ) x. 1 ) <_ ( ( ( B / A ) - 1 ) x. ( log ` A ) ) ) )
71	44, 32, 30, 69, 70	syl112anc 1330	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 <_ ( log ` A ) <-> ( ( ( B / A ) - 1 ) x. 1 ) <_ ( ( ( B / A ) - 1 ) x. ( log ` A ) ) ) )
72	66, 71	mpbid 222	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. 1 ) <_ ( ( ( B / A ) - 1 ) x. ( log ` A ) ) )
73	58, 72	eqbrtrrd 4677	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) - 1 ) <_ ( ( ( B / A ) - 1 ) x. ( log ` A ) ) )
74	26, 30, 33, 56, 73	ltletrd 10197	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` ( B / A ) ) < ( ( ( B / A ) - 1 ) x. ( log ` A ) ) )
75		relogdiv 24339	. . . . 5 |- ( ( B e. RR+ /\ A e. RR+ ) -> ( log ` ( B / A ) ) = ( ( log ` B ) - ( log ` A ) ) )
76	17, 23, 75	syl2anc 693	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` ( B / A ) ) = ( ( log ` B ) - ( log ` A ) ) )
77		1cnd 10056	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 1 e. CC )
78	32	recnd 10068	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` A ) e. CC )
79	37, 77, 78	subdird 10487	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. ( log ` A ) ) = ( ( ( B / A ) x. ( log ` A ) ) - ( 1 x. ( log ` A ) ) ) )
80	1	recnd 10068	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> B e. CC )
81	23	rpne0d 11877	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A =/= 0 )
82	80, 41, 78, 81	div32d 10824	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) x. ( log ` A ) ) = ( B x. ( ( log ` A ) / A ) ) )
83	78	mulid2d 10058	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 x. ( log ` A ) ) = ( log ` A ) )
84	82, 83	oveq12d 6668	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) x. ( log ` A ) ) - ( 1 x. ( log ` A ) ) ) = ( ( B x. ( ( log ` A ) / A ) ) - ( log ` A ) ) )
85	79, 84	eqtrd 2656	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. ( log ` A ) ) = ( ( B x. ( ( log ` A ) / A ) ) - ( log ` A ) ) )
86	74, 76, 85	3brtr3d 4684	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` B ) - ( log ` A ) ) < ( ( B x. ( ( log ` A ) / A ) ) - ( log ` A ) ) )
87		relogcl 24322	. . . . 5 |- ( B e. RR+ -> ( log ` B ) e. RR )
88	17, 87	syl 17	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` B ) e. RR )
89	32, 23	rerpdivcld 11903	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` A ) / A ) e. RR )
90	1, 89	remulcld 10070	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( B x. ( ( log ` A ) / A ) ) e. RR )
91	88, 90, 32	ltsub1d 10636	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` B ) < ( B x. ( ( log ` A ) / A ) ) <-> ( ( log ` B ) - ( log ` A ) ) < ( ( B x. ( ( log ` A ) / A ) ) - ( log ` A ) ) ) )
92	86, 91	mpbird 247	. 2 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` B ) < ( B x. ( ( log ` A ) / A ) ) )
93	88, 89, 17	ltdivmuld 11923	. 2 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( log ` B ) / B ) < ( ( log ` A ) / A ) <-> ( log ` B ) < ( B x. ( ( log ` A ) / A ) ) ) )
94	92, 93	mpbird 247	1 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` B ) / B ) < ( ( log ` A ) / A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   RR+crp 11832   expce 14792   _eceu 14793   logclog 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

This theorem is referenced by:  logdivlt  24367