logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 15611

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 1027	. . . . . . . 8 $/\$ $/\$ $/\$
2		simpl3 1028	. . . . . . . . . 10 $/\$ $/\$ $/\$
3		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$
4		ere 12236	. . . . . . . . . . 11
5		simpl1 1026	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8268	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1378	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 12347	. . . . . . . . . 10
10		0re 8179	. . . . . . . . . . 11
11		lttr 8253	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	10, 4, 1, 11	mp3an12i 1377	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	9, 12	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
14	8, 13	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
15	1, 14	elrpd 9928	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8269	. . . . . . . . . . 11 $/\$ $/\$ $/\$
17	10, 4, 5, 16	mp3an12i 1377	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	9, 17	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
19	2, 18	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
20	5, 19	elrpd 9928	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9949	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 15592	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9963	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8178	. . . . . 6
26		resubcl 8443	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 15592	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8210	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 15593	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 8125	. . . . . . . . 9
34	24	recnd 8208	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8387	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2267	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8208	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 8198	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4110	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8194	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 9054	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1277	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9927	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 12262	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4110	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 15505	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8208	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8196	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 12215	. . . . . . . . 9
57		reeflog 15593	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4116	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4124	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 15506	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8635	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 9037	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1277	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4112	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8603	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15600	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8195	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8208	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8594	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8208	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9937	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8994	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 8198	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 6036	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2264	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4119	. . 3 $/\$ $/\$ $/\$
84		relogcl 15592	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9963	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8210	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8734	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9983	. 2 $/\$ $/\$ $/\$
91	89, 90	mpbird 167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1004

wceq 1397

wcel 2202 class class class wbr 4088

cfv 5326 (class class class)co 6018

cc 8030

cr 8031

cc0 8032

c1 8033

caddc 8035

cmul 8037

clt 8214

cle 8215

cmin 8350

cdiv 8852

crp 9888

ce 12208

ceu 12209

clog 15586

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-pre-suploc 8153 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-ico 10129 df-icc 10130 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11380 df-cj 11407 df-re 11408 df-im 11409 df-rsqrt 11563 df-abs 11564 df-clim 11844 df-sumdc 11919 df-ef 12214 df-e 12215 df-rest 13329 df-topgen 13348 df-psmet 14563 df-xmet 14564 df-met 14565 df-bl 14566 df-mopn 14567 df-top 14728 df-topon 14741 df-bases 14773 df-ntr 14826 df-cn 14918 df-cnp 14919 df-tx 14983 df-cncf 15301 df-limced 15386 df-dvap 15387 df-relog 15588

This theorem is referenced by: (None)

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