logdivlti - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > logdivlti		Unicode version

Theorem logdivlti 15858

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 12381	. . . . . . . . . . 11
5		simpl1 1027	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8378	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1379	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 12492	. . . . . . . . . 10
10		0re 8290	. . . . . . . . . . 11
11		lttr 8363	. . . . . . . . . . 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 10044	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8379	. . . . . . . . . . 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 10044	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 10065	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 15839	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 10079	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8289	. . . . . 6
26		resubcl 8553	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 15839	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8320	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 15840	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 8236	. . . . . . . . 9
34	24	recnd 8318	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8497	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2270	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8318	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mullidd 8308	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4136	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8305	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 9165	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 10043	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 12407	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4136	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 15752	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8318	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8307	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 12360	. . . . . . . . 9
57		reeflog 15840	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4142	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4150	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 15753	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8746	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 9148	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4138	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8714	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15847	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8306	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8318	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8705	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8318	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 10053	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 9105	. . . . . 6 $/\$ $/\$ $/\$
80	75	mullidd 8308	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 6076	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2267	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4145	. . 3 $/\$ $/\$ $/\$
84		relogcl 15839	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 10079	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8320	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8845	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 10099	. 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 4114

cfv 5357 (class class class)co 6058

cc 8141

cr 8142

cc0 8143

c1 8144

caddc 8146

cmul 8148

clt 8324

cle 8325

cmin 8460

cdiv 8963

crp 10004

ce 12353

ceu 12354

clog 15833

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 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 ax-pre-suploc 8264 ax-addf 8265 ax-mulf 8266

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 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-disj 4091 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-of 6275 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-map 6897 df-pm 6898 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-xneg 10124 df-xadd 10125 df-ioo 10244 df-ico 10246 df-icc 10247 df-fz 10362 df-fzo 10499 df-seqfrec 10834 df-exp 10925 df-fac 11113 df-bc 11135 df-ihash 11164 df-shft 11525 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-sumdc 12064 df-ef 12359 df-e 12360 df-rest 13538 df-topgen 13557 df-psmet 14803 df-xmet 14804 df-met 14805 df-bl 14806 df-mopn 14807 df-top 14975 df-topon 14988 df-bases 15020 df-ntr 15073 df-cn 15165 df-cnp 15166 df-tx 15230 df-cncf 15548 df-limced 15633 df-dvap 15634 df-relog 15835

This theorem is referenced by: (None)

W3C validator