logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 15057

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 11816	. . . . . . . . . . 11
5		simpl1 1002	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8110	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1353	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 11927	. . . . . . . . . 10
10		0re 8021	. . . . . . . . . . 11
11		lttr 8095	. . . . . . . . . . 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 9762	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8111	. . . . . . . . . . 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 9762	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9783	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 15038	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9797	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8020	. . . . . 6
26		resubcl 8285	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 15038	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8052	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 15039	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 7967	. . . . . . . . 9
34	24	recnd 8050	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8229	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2229	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8050	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 8040	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4052	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8036	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 8895	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1253	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9761	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 11842	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4052	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 14951	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8050	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8038	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 11795	. . . . . . . . 9
57		reeflog 15039	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4058	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4066	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 14952	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8476	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 8878	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1253	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4054	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8444	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15046	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8037	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8050	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8436	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8050	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9771	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8835	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 8040	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 5937	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2226	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4061	. . 3 $/\$ $/\$ $/\$
84		relogcl 15038	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9797	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8052	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8575	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9817	. 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 4030

cfv 5255 (class class class)co 5919

cc 7872

cr 7873

cc0 7874

c1 7875

caddc 7877

cmul 7879

clt 8056

cle 8057

cmin 8192

cdiv 8693

crp 9722

ce 11788

ceu 11789

clog 15032

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 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994 ax-pre-suploc 7995 ax-addf 7996 ax-mulf 7997

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 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-disj 4008 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-of 6132 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-oadd 6475 df-er 6589 df-map 6706 df-pm 6707 df-en 6797 df-dom 6798 df-fin 6799 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-xneg 9841 df-xadd 9842 df-ioo 9961 df-ico 9963 df-icc 9964 df-fz 10078 df-fzo 10212 df-seqfrec 10522 df-exp 10613 df-fac 10800 df-bc 10822 df-ihash 10850 df-shft 10962 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-sumdc 11500 df-ef 11794 df-e 11795 df-rest 12855 df-topgen 12874 df-psmet 14042 df-xmet 14043 df-met 14044 df-bl 14045 df-mopn 14046 df-top 14177 df-topon 14190 df-bases 14222 df-ntr 14275 df-cn 14367 df-cnp 14368 df-tx 14432 df-cncf 14750 df-limced 14835 df-dvap 14836 df-relog 15034

This theorem is referenced by: (None)

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