logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 13969

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 1001	. . . . . . . 8 $/\$ $/\$ $/\$
2		simpl3 1002	. . . . . . . . . 10 $/\$ $/\$ $/\$
3		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$
4		ere 11662	. . . . . . . . . . 11
5		simpl1 1000	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8036	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1342	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 11772	. . . . . . . . . 10
10		0re 7948	. . . . . . . . . . 11
11		lttr 8021	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	10, 4, 1, 11	mp3an12i 1341	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	9, 12	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
14	8, 13	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
15	1, 14	elrpd 9680	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8037	. . . . . . . . . . 11 $/\$ $/\$ $/\$
17	10, 4, 5, 16	mp3an12i 1341	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	9, 17	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
19	2, 18	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
20	5, 19	elrpd 9680	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9701	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 13950	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9715	. . . . . 6 $/\$ $/\$ $/\$
25		1re 7947	. . . . . 6
26		resubcl 8211	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 13950	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 7978	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 13951	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 7895	. . . . . . . . 9
34	24	recnd 7976	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8155	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2213	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 7976	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 7966	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4022	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 7963	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 8820	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1242	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9679	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 11688	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4022	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 13863	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 7976	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulid1d 7965	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 11641	. . . . . . . . 9
57		reeflog 13951	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4028	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4036	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 13864	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8402	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 8803	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1242	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4024	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8370	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 13958	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 7964	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 7976	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8362	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 7976	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9689	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8760	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 7966	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 5887	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2210	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4031	. . 3 $/\$ $/\$ $/\$
84		relogcl 13950	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9715	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 7978	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8501	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9735	. 2 $/\$ $/\$ $/\$
91	89, 90	mpbird 167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 978

wceq 1353

wcel 2148 class class class wbr 4000

cfv 5212 (class class class)co 5869

cc 7800

cr 7801

cc0 7802

c1 7803

caddc 7805

cmul 7807

clt 7982

cle 7983

cmin 8118

cdiv 8618

crp 9640

ce 11634

ceu 11635

clog 13944

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 ax-pre-suploc 7923 ax-addf 7924 ax-mulf 7925

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-disj 3978 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-of 6077 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-oadd 6415 df-er 6529 df-map 6644 df-pm 6645 df-en 6735 df-dom 6736 df-fin 6737 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-xneg 9759 df-xadd 9760 df-ioo 9879 df-ico 9881 df-icc 9882 df-fz 9996 df-fzo 10129 df-seqfrec 10432 df-exp 10506 df-fac 10690 df-bc 10712 df-ihash 10740 df-shft 10808 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-sumdc 11346 df-ef 11640 df-e 11641 df-rest 12638 df-topgen 12657 df-psmet 13154 df-xmet 13155 df-met 13156 df-bl 13157 df-mopn 13158 df-top 13163 df-topon 13176 df-bases 13208 df-ntr 13263 df-cn 13355 df-cnp 13356 df-tx 13420 df-cncf 13725 df-limced 13792 df-dvap 13793 df-relog 13946

This theorem is referenced by: (None)

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