Theorem logdivlti 15986

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 1032	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> B e. RR )
2		simpl3 1033	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> _e <_ A )
3		simpr 110	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A < B )
4		ere 12439	. . . . . . . . . . 11 |- _e e. RR
5		simpl1 1031	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A e. RR )
6		lelttr 8414	. . . . . . . . . . 11 |- ( ( _e e. RR /\ A e. RR /\ B e. RR ) -> ( ( _e <_ A /\ A < B ) -> _e < B ) )
7	4, 5, 1, 6	mp3an2i 1383	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( _e <_ A /\ A < B ) -> _e < B ) )
8	2, 3, 7	mp2and 437	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> _e < B )
9		epos 12550	. . . . . . . . . 10 |- 0 < _e
10		0re 8326	. . . . . . . . . . 11 |- 0 e. RR
11		lttr 8399	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ _e e. RR /\ B e. RR ) -> ( ( 0 < _e /\ _e < B ) -> 0 < B ) )
12	10, 4, 1, 11	mp3an12i 1382	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( 0 < _e /\ _e < B ) -> 0 < B ) )
13	9, 12	mpani 434	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( _e < B -> 0 < B ) )
14	8, 13	mpd 13	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 0 < B )
15	1, 14	elrpd 10096	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> B e. RR+ )
16		ltletr 8415	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ _e e. RR /\ A e. RR ) -> ( ( 0 < _e /\ _e <_ A ) -> 0 < A ) )
17	10, 4, 5, 16	mp3an12i 1382	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( 0 < _e /\ _e <_ A ) -> 0 < A ) )
18	9, 17	mpani 434	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( _e <_ A -> 0 < A ) )
19	2, 18	mpd 13	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 0 < A )
20	5, 19	elrpd 10096	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A e. RR+ )
21	15, 20	rpdivcld 10117	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( B / A ) e. RR+ )
22		relogcl 15966	. . . . . 6 |- ( ( B / A ) e. RR+ -> ( log ` ( B / A ) ) e. RR )
23	21, 22	syl 14	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` ( B / A ) ) e. RR )
24	1, 20	rerpdivcld 10131	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( B / A ) e. RR )
25		1re 8325	. . . . . 6 |- 1 e. RR
26		resubcl 8590	. . . . . 6 |- ( ( ( B / A ) e. RR /\ 1 e. RR ) -> ( ( B / A ) - 1 ) e. RR )
27	24, 25, 26	sylancl 417	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) - 1 ) e. RR )
28		relogcl 15966	. . . . . . 7 |- ( A e. RR+ -> ( log ` A ) e. RR )
29	20, 28	syl 14	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` A ) e. RR )
30	27, 29	remulcld 8356	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. ( log ` A ) ) e. RR )
31		reeflog 15967	. . . . . . . . 9 |- ( ( B / A ) e. RR+ -> ( exp ` ( log ` ( B / A ) ) ) = ( B / A ) )
32	21, 31	syl 14	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` ( log ` ( B / A ) ) ) = ( B / A ) )
33		ax-1cn 8272	. . . . . . . . 9 |- 1 e. CC
34	24	recnd 8354	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( B / A ) e. CC )
35		pncan3 8534	. . . . . . . . 9 |- ( ( 1 e. CC /\ ( B / A ) e. CC ) -> ( 1 + ( ( B / A ) - 1 ) ) = ( B / A ) )
36	33, 34, 35	sylancr 418	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 + ( ( B / A ) - 1 ) ) = ( B / A ) )
37	32, 36	eqtr4d 2274	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` ( log ` ( B / A ) ) ) = ( 1 + ( ( B / A ) - 1 ) ) )
38	5	recnd 8354	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A e. CC )
39	38	mullidd 8344	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 x. A ) = A )
40	39, 3	eqbrtrd 4152	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 x. A ) < B )
41		1red 8341	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 1 e. RR )
42		ltmuldiv 9205	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
43	41, 1, 5, 19, 42	syl112anc 1282	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
44	40, 43	mpbid 147	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 1 < ( B / A ) )
45		difrp 10095	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( B / A ) e. RR ) -> ( 1 < ( B / A ) <-> ( ( B / A ) - 1 ) e. RR+ ) )
46	25, 24, 45	sylancr 418	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 < ( B / A ) <-> ( ( B / A ) - 1 ) e. RR+ ) )
47	44, 46	mpbid 147	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) - 1 ) e. RR+ )
48		efgt1p 12465	. . . . . . . 8 |- ( ( ( B / A ) - 1 ) e. RR+ -> ( 1 + ( ( B / A ) - 1 ) ) < ( exp ` ( ( B / A ) - 1 ) ) )
49	47, 48	syl 14	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 + ( ( B / A ) - 1 ) ) < ( exp ` ( ( B / A ) - 1 ) ) )
50	37, 49	eqbrtrd 4152	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` ( log ` ( B / A ) ) ) < ( exp ` ( ( B / A ) - 1 ) ) )
51		eflt 15878	. . . . . . 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 ) ) ) )
52	23, 27, 51	syl2anc 415	. . . . . 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 ) ) ) )
53	50, 52	mpbird 167	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` ( B / A ) ) < ( ( B / A ) - 1 ) )
54	27	recnd 8354	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) - 1 ) e. CC )
55	54	mulridd 8343	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. 1 ) = ( ( B / A ) - 1 ) )
56		df-e 12418	. . . . . . . . 9 |- _e = ( exp ` 1 )
57		reeflog 15967	. . . . . . . . . . 11 |- ( A e. RR+ -> ( exp ` ( log ` A ) ) = A )
58	20, 57	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` ( log ` A ) ) = A )
59	2, 58	breqtrrd 4158	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> _e <_ ( exp ` ( log ` A ) ) )
60	56, 59	eqbrtrrid 4166	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( exp ` 1 ) <_ ( exp ` ( log ` A ) ) )
61		efle 15879	. . . . . . . . 9 |- ( ( 1 e. RR /\ ( log ` A ) e. RR ) -> ( 1 <_ ( log ` A ) <-> ( exp ` 1 ) <_ ( exp ` ( log ` A ) ) ) )
62	25, 29, 61	sylancr 418	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 <_ ( log ` A ) <-> ( exp ` 1 ) <_ ( exp ` ( log ` A ) ) ) )
63	60, 62	mpbird 167	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 1 <_ ( log ` A ) )
64		posdif 8783	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( B / A ) e. RR ) -> ( 1 < ( B / A ) <-> 0 < ( ( B / A ) - 1 ) ) )
65	25, 24, 64	sylancr 418	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 < ( B / A ) <-> 0 < ( ( B / A ) - 1 ) ) )
66	44, 65	mpbid 147	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 0 < ( ( B / A ) - 1 ) )
67		lemul2 9188	. . . . . . . 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 ) ) ) )
68	41, 29, 27, 66, 67	syl112anc 1282	. . . . . . 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 ) ) ) )
69	63, 68	mpbid 147	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. 1 ) <_ ( ( ( B / A ) - 1 ) x. ( log ` A ) ) )
70	55, 69	eqbrtrrd 4154	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) - 1 ) <_ ( ( ( B / A ) - 1 ) x. ( log ` A ) ) )
71	23, 27, 30, 53, 70	ltletrd 8751	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` ( B / A ) ) < ( ( ( B / A ) - 1 ) x. ( log ` A ) ) )
72		relogdiv 15975	. . . . 5 |- ( ( B e. RR+ /\ A e. RR+ ) -> ( log ` ( B / A ) ) = ( ( log ` B ) - ( log ` A ) ) )
73	15, 20, 72	syl2anc 415	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` ( B / A ) ) = ( ( log ` B ) - ( log ` A ) ) )
74		1cnd 8342	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> 1 e. CC )
75	29	recnd 8354	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` A ) e. CC )
76	34, 74, 75	subdird 8742	. . . . 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 ) ) ) )
77	1	recnd 8354	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> B e. CC )
78	20	rpap0d 10105	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> A # 0 )
79	77, 38, 75, 78	div32apd 9145	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( B / A ) x. ( log ` A ) ) = ( B x. ( ( log ` A ) / A ) ) )
80	75	mullidd 8344	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( 1 x. ( log ` A ) ) = ( log ` A ) )
81	79, 80	oveq12d 6103	. . . . 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 ) ) )
82	76, 81	eqtrd 2271	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( B / A ) - 1 ) x. ( log ` A ) ) = ( ( B x. ( ( log ` A ) / A ) ) - ( log ` A ) ) )
83	71, 73, 82	3brtr3d 4161	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` B ) - ( log ` A ) ) < ( ( B x. ( ( log ` A ) / A ) ) - ( log ` A ) ) )
84		relogcl 15966	. . . . 5 |- ( B e. RR+ -> ( log ` B ) e. RR )
85	15, 84	syl 14	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` B ) e. RR )
86	29, 20	rerpdivcld 10131	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` A ) / A ) e. RR )
87	1, 86	remulcld 8356	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( B x. ( ( log ` A ) / A ) ) e. RR )
88	85, 87, 29	ltsub1d 8882	. . 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 ) ) ) )
89	83, 88	mpbird 167	. 2 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( log ` B ) < ( B x. ( ( log ` A ) / A ) ) )
90	85, 86, 15	ltdivmuld 10151	. 2 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( ( log ` B ) / B ) < ( ( log ` A ) / A ) <-> ( log ` B ) < ( B x. ( ( log ` A ) / A ) ) ) )
91	89, 90	mpbird 167	1 |- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` B ) / B ) < ( ( log ` A ) / A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   - cmin 8497   / cdiv 9003   RR+crp 10056   expce 12411   _eceu 12412   logclog 15960

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-xneg 10176  df-xadd 10177  df-ioo 10296  df-ico 10298  df-icc 10299  df-fz 10414  df-fzo 10552  df-seqfrec 10887  df-exp 10978  df-fac 11166  df-bc 11188  df-ihash 11217  df-shft 11582  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-clim 12047  df-sumdc 12122  df-ef 12417  df-e 12418  df-rest 13597  df-topgen 13616  df-psmet 14882  df-xmet 14883  df-met 14884  df-bl 14885  df-mopn 14886  df-top 15101  df-topon 15114  df-bases 15146  df-ntr 15199  df-cn 15291  df-cnp 15292  df-tx 15356  df-cncf 15674  df-limced 15759  df-dvap 15760  df-relog 15962

This theorem is used by:  logdivlt  15999