logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 15016

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 1003	. . . . . . . 8 $/\$ $/\$ $/\$
2		simpl3 1004	. . . . . . . . . 10 $/\$ $/\$ $/\$
3		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$
4		ere 11813	. . . . . . . . . . 11
5		simpl1 1002	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8108	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1353	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 11924	. . . . . . . . . 10
10		0re 8019	. . . . . . . . . . 11
11		lttr 8093	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	10, 4, 1, 11	mp3an12i 1352	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	9, 12	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
14	8, 13	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
15	1, 14	elrpd 9759	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8109	. . . . . . . . . . 11 $/\$ $/\$ $/\$
17	10, 4, 5, 16	mp3an12i 1352	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	9, 17	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
19	2, 18	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
20	5, 19	elrpd 9759	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9780	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 14997	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9794	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8018	. . . . . 6
26		resubcl 8283	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 14997	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8050	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 14998	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 7965	. . . . . . . . 9
34	24	recnd 8048	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8227	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2229	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8048	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 8038	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4051	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8034	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 8893	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1253	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9758	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 11839	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4051	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 14910	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8048	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8036	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 11792	. . . . . . . . 9
57		reeflog 14998	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4057	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4065	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 14911	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8474	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 8876	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1253	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4053	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8442	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15005	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8035	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8048	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8434	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8048	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9768	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8833	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 8038	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 5936	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2226	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4060	. . 3 $/\$ $/\$ $/\$
84		relogcl 14997	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9794	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8050	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8573	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9814	. 2 $/\$ $/\$ $/\$
91	89, 90	mpbird 167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

wcel 2164 class class class wbr 4029

cfv 5254 (class class class)co 5918

cc 7870

cr 7871

cc0 7872

c1 7873

caddc 7875

cmul 7877

clt 8054

cle 8055

cmin 8190

cdiv 8691

crp 9719

ce 11785

ceu 11786

clog 14991

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 ax-pre-suploc 7993 ax-addf 7994 ax-mulf 7995

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-disj 4007 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-of 6130 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-map 6704 df-pm 6705 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-xneg 9838 df-xadd 9839 df-ioo 9958 df-ico 9960 df-icc 9961 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-fac 10797 df-bc 10819 df-ihash 10847 df-shft 10959 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497 df-ef 11791 df-e 11792 df-rest 12852 df-topgen 12871 df-psmet 14039 df-xmet 14040 df-met 14041 df-bl 14042 df-mopn 14043 df-top 14166 df-topon 14179 df-bases 14211 df-ntr 14264 df-cn 14356 df-cnp 14357 df-tx 14421 df-cncf 14726 df-limced 14810 df-dvap 14811 df-relog 14993

This theorem is referenced by: (None)

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