logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 15672

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 12292	. . . . . . . . . . 11
5		simpl1 1027	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8311	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1379	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 12403	. . . . . . . . . 10
10		0re 8222	. . . . . . . . . . 11
11		lttr 8296	. . . . . . . . . . 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 9971	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8312	. . . . . . . . . . 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 9971	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9992	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 15653	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 10006	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8221	. . . . . 6
26		resubcl 8486	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 15653	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8253	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 15654	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 8168	. . . . . . . . 9
34	24	recnd 8251	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8430	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2267	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8251	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 8241	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4115	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8237	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 9097	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9970	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 12318	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4115	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 15566	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8251	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8239	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 12271	. . . . . . . . 9
57		reeflog 15654	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4121	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4129	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 15567	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8678	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 9080	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4117	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8646	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15661	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8238	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8251	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8637	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8251	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9980	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 9037	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 8241	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 6046	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2264	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4124	. . 3 $/\$ $/\$ $/\$
84		relogcl 15653	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 10006	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8253	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8777	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 10026	. 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 2202 class class class wbr 4093

cfv 5333 (class class class)co 6028

cc 8073

cr 8074

cc0 8075

c1 8076

caddc 8078

cmul 8080

clt 8257

cle 8258

cmin 8393

cdiv 8895

crp 9931

ce 12264

ceu 12265

clog 15647

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195 ax-pre-suploc 8196 ax-addf 8197 ax-mulf 8198

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-disj 4070 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-of 6244 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-oadd 6629 df-er 6745 df-map 6862 df-pm 6863 df-en 6953 df-dom 6954 df-fin 6955 df-sup 7226 df-inf 7227 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-n0 9446 df-z 9523 df-uz 9799 df-q 9897 df-rp 9932 df-xneg 10050 df-xadd 10051 df-ioo 10170 df-ico 10172 df-icc 10173 df-fz 10287 df-fzo 10421 df-seqfrec 10754 df-exp 10845 df-fac 11032 df-bc 11054 df-ihash 11082 df-shft 11436 df-cj 11463 df-re 11464 df-im 11465 df-rsqrt 11619 df-abs 11620 df-clim 11900 df-sumdc 11975 df-ef 12270 df-e 12271 df-rest 13385 df-topgen 13404 df-psmet 14619 df-xmet 14620 df-met 14621 df-bl 14622 df-mopn 14623 df-top 14789 df-topon 14802 df-bases 14834 df-ntr 14887 df-cn 14979 df-cnp 14980 df-tx 15044 df-cncf 15362 df-limced 15447 df-dvap 15448 df-relog 15649

This theorem is referenced by: (None)

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