logdivlti - Intuitionistic Logic Explorer

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

Theorem logdivlti 13442

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 991	. . . . . . . 8 $/\$ $/\$ $/\$
2		simpl3 992	. . . . . . . . . 10 $/\$ $/\$ $/\$
3		simpr 109	. . . . . . . . . 10 $/\$ $/\$ $/\$
4		ere 11611	. . . . . . . . . . 11
5		simpl1 990	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 7987	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1332	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 430	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 11721	. . . . . . . . . 10
10		0re 7899	. . . . . . . . . . 11
11		lttr 7972	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	10, 4, 1, 11	mp3an12i 1331	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	9, 12	mpani 427	. . . . . . . . 9 $/\$ $/\$ $/\$
14	8, 13	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
15	1, 14	elrpd 9629	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 7988	. . . . . . . . . . 11 $/\$ $/\$ $/\$
17	10, 4, 5, 16	mp3an12i 1331	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	9, 17	mpani 427	. . . . . . . . 9 $/\$ $/\$ $/\$
19	2, 18	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
20	5, 19	elrpd 9629	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9650	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 13423	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9664	. . . . . 6 $/\$ $/\$ $/\$
25		1re 7898	. . . . . 6
26		resubcl 8162	. . . . . 6 $/\$
27	24, 25, 26	sylancl 410	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 13423	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 7929	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 13424	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 7846	. . . . . . . . 9
34	24	recnd 7927	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8106	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 411	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2201	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 7927	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 7917	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4004	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 7914	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 8769	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1232	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 146	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9628	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 411	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 11637	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4004	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 13336	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 409	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 166	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 7927	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulid1d 7916	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 11590	. . . . . . . . 9
57		reeflog 13424	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4010	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4018	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 13337	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 411	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 166	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8353	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 411	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 8752	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1232	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4006	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8321	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 13431	. . . . 5 $/\$
73	15, 20, 72	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
74		1cnd 7915	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 7927	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8313	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 7927	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9638	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8710	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 7917	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 5860	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2198	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4013	. . 3 $/\$ $/\$ $/\$
84		relogcl 13423	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9664	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 7929	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8452	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 166	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9684	. 2 $/\$ $/\$ $/\$
91	89, 90	mpbird 166	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 968

wceq 1343

wcel 2136 class class class wbr 3982

cfv 5188 (class class class)co 5842

cc 7751

cr 7752

cc0 7753

c1 7754

caddc 7756

cmul 7758

clt 7933

cle 7934

cmin 8069

cdiv 8568

crp 9589

ce 11583

ceu 11584

clog 13417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-pre-suploc 7874 ax-addf 7875 ax-mulf 7876

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-disj 3960 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-map 6616 df-pm 6617 df-en 6707 df-dom 6708 df-fin 6709 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-ioo 9828 df-ico 9830 df-icc 9831 df-fz 9945 df-fzo 10078 df-seqfrec 10381 df-exp 10455 df-fac 10639 df-bc 10661 df-ihash 10689 df-shft 10757 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-sumdc 11295 df-ef 11589 df-e 11590 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-ntr 12736 df-cn 12828 df-cnp 12829 df-tx 12893 df-cncf 13198 df-limced 13265 df-dvap 13266 df-relog 13419

This theorem is referenced by: (None)

W3C validator