logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 15877

Description: The

function is strictly decreasing on the reals greater than

. (Contributed by Mario Carneiro, 14-Mar-2014.)

**Assertion**
Ref	Expression
logdivlti	$/\$ $/\$ $/\$

**Proof of Theorem logdivlti**
Step	Hyp	Ref	Expression
1		simpl2 1028	. . . . . . . 8 $/\$ $/\$ $/\$
2		simpl3 1029	. . . . . . . . . 10 $/\$ $/\$ $/\$
3		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$
4		ere 12386	. . . . . . . . . . 11
5		simpl1 1027	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8379	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1379	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 12497	. . . . . . . . . 10
10		0re 8291	. . . . . . . . . . 11
11		lttr 8364	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	10, 4, 1, 11	mp3an12i 1378	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	9, 12	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
14	8, 13	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
15	1, 14	elrpd 10048	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8380	. . . . . . . . . . 11 $/\$ $/\$ $/\$
17	10, 4, 5, 16	mp3an12i 1378	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	9, 17	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
19	2, 18	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
20	5, 19	elrpd 10048	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 10069	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 15858	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 10083	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8290	. . . . . 6
26		resubcl 8555	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 15858	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8321	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 15859	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 8237	. . . . . . . . 9
34	24	recnd 8319	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8499	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2270	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8319	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mullidd 8309	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4137	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8306	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 9169	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 10047	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 12412	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4137	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 15771	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8319	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8308	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 12365	. . . . . . . . 9
57		reeflog 15859	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4143	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4151	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 15772	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8748	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 9152	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4139	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8716	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15866	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8307	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8319	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8707	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8319	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 10057	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 9109	. . . . . 6 $/\$ $/\$ $/\$
80	75	mullidd 8309	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 6077	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2267	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4146	. . 3 $/\$ $/\$ $/\$
84		relogcl 15858	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 10083	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8321	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8847	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 10103	. 2 $/\$ $/\$ $/\$
91	89, 90	mpbird 167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1005

wceq 1398

wcel 2205 class class class wbr 4115

cfv 5358 (class class class)co 6059

cc 8142

cr 8143

cc0 8144

c1 8145

caddc 8147

cmul 8149

clt 8325

cle 8326

cmin 8462

cdiv 8967

crp 10008

ce 12358

ceu 12359

clog 15852

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4231 ax-sep 4234 ax-nul 4242 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-setind 4665 ax-iinf 4716 ax-cnex 8235 ax-resscn 8236 ax-1cn 8237 ax-1re 8238 ax-icn 8239 ax-addcl 8240 ax-addrcl 8241 ax-mulcl 8242 ax-mulrcl 8243 ax-addcom 8244 ax-mulcom 8245 ax-addass 8246 ax-mulass 8247 ax-distr 8248 ax-i2m1 8249 ax-0lt1 8250 ax-1rid 8251 ax-0id 8252 ax-rnegex 8253 ax-precex 8254 ax-cnre 8255 ax-pre-ltirr 8256 ax-pre-ltwlin 8257 ax-pre-lttrn 8258 ax-pre-apti 8259 ax-pre-ltadd 8260 ax-pre-mulgt0 8261 ax-pre-mulext 8262 ax-arch 8263 ax-caucvg 8264 ax-pre-suploc 8265 ax-addf 8266 ax-mulf 8267

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3626 df-pw 3677 df-sn 3701 df-pr 3702 df-op 3704 df-uni 3921 df-int 3956 df-iun 3999 df-disj 4092 df-br 4116 df-opab 4178 df-mpt 4179 df-tr 4215 df-id 4420 df-po 4423 df-iso 4424 df-iord 4493 df-on 4495 df-ilim 4496 df-suc 4498 df-iom 4719 df-xp 4761 df-rel 4762 df-cnv 4763 df-co 4764 df-dm 4765 df-rn 4766 df-res 4767 df-ima 4768 df-iota 5318 df-fun 5360 df-fn 5361 df-f 5362 df-f1 5363 df-fo 5364 df-f1o 5365 df-fv 5366 df-isom 5367 df-riota 6012 df-ov 6062 df-oprab 6063 df-mpo 6064 df-of 6276 df-1st 6348 df-2nd 6349 df-recs 6550 df-irdg 6615 df-frec 6636 df-1o 6661 df-oadd 6665 df-er 6781 df-map 6898 df-pm 6899 df-en 6990 df-dom 6991 df-fin 6992 df-sup 7289 df-inf 7290 df-pnf 8327 df-mnf 8328 df-xr 8329 df-ltxr 8330 df-le 8331 df-sub 8464 df-neg 8465 df-reap 8868 df-ap 8875 df-div 8968 df-inn 9259 df-2 9317 df-3 9318 df-4 9319 df-n0 9518 df-z 9599 df-uz 9876 df-q 9974 df-rp 10009 df-xneg 10128 df-xadd 10129 df-ioo 10248 df-ico 10250 df-icc 10251 df-fz 10366 df-fzo 10503 df-seqfrec 10838 df-exp 10929 df-fac 11117 df-bc 11139 df-ihash 11168 df-shft 11529 df-cj 11556 df-re 11557 df-im 11558 df-rsqrt 11713 df-abs 11714 df-clim 11994 df-sumdc 12069 df-ef 12364 df-e 12365 df-rest 13543 df-topgen 13562 df-psmet 14822 df-xmet 14823 df-met 14824 df-bl 14825 df-mopn 14826 df-top 14994 df-topon 15007 df-bases 15039 df-ntr 15092 df-cn 15184 df-cnp 15185 df-tx 15249 df-cncf 15567 df-limced 15652 df-dvap 15653 df-relog 15854

This theorem is referenced by: (None)

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