logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 15738

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 12352	. . . . . . . . . . 11
5		simpl1 1027	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8361	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1379	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 12463	. . . . . . . . . 10
10		0re 8273	. . . . . . . . . . 11
11		lttr 8346	. . . . . . . . . . 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 10025	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8362	. . . . . . . . . . 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 10025	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 10046	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 15719	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 10060	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8272	. . . . . 6
26		resubcl 8536	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 15719	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8303	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 15720	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 8219	. . . . . . . . 9
34	24	recnd 8301	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8480	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2268	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8301	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mullidd 8291	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4130	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8288	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 9147	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 10024	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 12378	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4130	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 15632	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8301	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8290	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 12331	. . . . . . . . 9
57		reeflog 15720	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4136	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4144	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 15633	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8728	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 9130	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4132	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8696	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15727	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8289	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8301	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8687	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8301	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 10034	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 9087	. . . . . 6 $/\$ $/\$ $/\$
80	75	mullidd 8291	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 6067	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2265	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4139	. . 3 $/\$ $/\$ $/\$
84		relogcl 15719	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 10060	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8303	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8827	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 10080	. 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 2203 class class class wbr 4108

cfv 5351 (class class class)co 6049

cc 8124

cr 8125

cc0 8126

c1 8127

caddc 8129

cmul 8131

clt 8307

cle 8308

cmin 8443

cdiv 8945

crp 9985

ce 12324

ceu 12325

clog 15713

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-pre-mulext 8244 ax-arch 8245 ax-caucvg 8246 ax-pre-suploc 8247 ax-addf 8248 ax-mulf 8249

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 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-disj 4085 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-of 6265 df-1st 6333 df-2nd 6334 df-recs 6535 df-irdg 6600 df-frec 6621 df-1o 6646 df-oadd 6650 df-er 6766 df-map 6883 df-pm 6884 df-en 6975 df-dom 6976 df-fin 6977 df-sup 7274 df-inf 7275 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-ap 8855 df-div 8946 df-inn 9237 df-2 9295 df-3 9296 df-4 9297 df-n0 9496 df-z 9577 df-uz 9853 df-q 9951 df-rp 9986 df-xneg 10104 df-xadd 10105 df-ioo 10224 df-ico 10226 df-icc 10227 df-fz 10342 df-fzo 10476 df-seqfrec 10809 df-exp 10900 df-fac 11087 df-bc 11109 df-ihash 11137 df-shft 11496 df-cj 11523 df-re 11524 df-im 11525 df-rsqrt 11679 df-abs 11680 df-clim 11960 df-sumdc 12035 df-ef 12330 df-e 12331 df-rest 13446 df-topgen 13465 df-psmet 14683 df-xmet 14684 df-met 14685 df-bl 14686 df-mopn 14687 df-top 14855 df-topon 14868 df-bases 14900 df-ntr 14953 df-cn 15045 df-cnp 15046 df-tx 15110 df-cncf 15428 df-limced 15513 df-dvap 15514 df-relog 15715

This theorem is referenced by: (None)

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