logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 15563

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 1025	. . . . . . . 8 $/\$ $/\$ $/\$
2		simpl3 1026	. . . . . . . . . 10 $/\$ $/\$ $/\$
3		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$
4		ere 12189	. . . . . . . . . . 11
5		simpl1 1024	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8243	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1376	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 12300	. . . . . . . . . 10
10		0re 8154	. . . . . . . . . . 11
11		lttr 8228	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	10, 4, 1, 11	mp3an12i 1375	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	9, 12	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
14	8, 13	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
15	1, 14	elrpd 9897	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8244	. . . . . . . . . . 11 $/\$ $/\$ $/\$
17	10, 4, 5, 16	mp3an12i 1375	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	9, 17	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
19	2, 18	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
20	5, 19	elrpd 9897	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9918	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 15544	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9932	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8153	. . . . . 6
26		resubcl 8418	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 15544	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8185	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 15545	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 8100	. . . . . . . . 9
34	24	recnd 8183	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8362	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2265	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8183	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 8173	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4105	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8169	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 9029	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1275	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9896	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 12215	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4105	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 15457	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8183	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8171	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 12168	. . . . . . . . 9
57		reeflog 15545	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4111	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4119	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 15458	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8610	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 9012	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1275	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4107	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8578	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15552	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8170	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8183	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8569	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8183	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9906	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8969	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 8173	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 6025	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2262	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4114	. . 3 $/\$ $/\$ $/\$
84		relogcl 15544	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9932	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8185	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8709	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9952	. 2 $/\$ $/\$ $/\$
91	89, 90	mpbird 167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1002

wceq 1395

wcel 2200 class class class wbr 4083

cfv 5318 (class class class)co 6007

cc 8005

cr 8006

cc0 8007

c1 8008

caddc 8010

cmul 8012

clt 8189

cle 8190

cmin 8325

cdiv 8827

crp 9857

ce 12161

ceu 12162

clog 15538

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127 ax-pre-suploc 8128 ax-addf 8129 ax-mulf 8130

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-disj 4060 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-of 6224 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-map 6805 df-pm 6806 df-en 6896 df-dom 6897 df-fin 6898 df-sup 7159 df-inf 7160 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-xneg 9976 df-xadd 9977 df-ioo 10096 df-ico 10098 df-icc 10099 df-fz 10213 df-fzo 10347 df-seqfrec 10678 df-exp 10769 df-fac 10956 df-bc 10978 df-ihash 11006 df-shft 11334 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-clim 11798 df-sumdc 11873 df-ef 12167 df-e 12168 df-rest 13282 df-topgen 13301 df-psmet 14515 df-xmet 14516 df-met 14517 df-bl 14518 df-mopn 14519 df-top 14680 df-topon 14693 df-bases 14725 df-ntr 14778 df-cn 14870 df-cnp 14871 df-tx 14935 df-cncf 15253 df-limced 15338 df-dvap 15339 df-relog 15540

This theorem is referenced by: (None)

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