logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 13596

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 996	. . . . . . . 8 $/\$ $/\$ $/\$
2		simpl3 997	. . . . . . . . . 10 $/\$ $/\$ $/\$
3		simpr 109	. . . . . . . . . 10 $/\$ $/\$ $/\$
4		ere 11633	. . . . . . . . . . 11
5		simpl1 995	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8008	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1337	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 431	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 11743	. . . . . . . . . 10
10		0re 7920	. . . . . . . . . . 11
11		lttr 7993	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	10, 4, 1, 11	mp3an12i 1336	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	9, 12	mpani 428	. . . . . . . . 9 $/\$ $/\$ $/\$
14	8, 13	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
15	1, 14	elrpd 9650	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8009	. . . . . . . . . . 11 $/\$ $/\$ $/\$
17	10, 4, 5, 16	mp3an12i 1336	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	9, 17	mpani 428	. . . . . . . . 9 $/\$ $/\$ $/\$
19	2, 18	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
20	5, 19	elrpd 9650	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9671	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 13577	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9685	. . . . . 6 $/\$ $/\$ $/\$
25		1re 7919	. . . . . 6
26		resubcl 8183	. . . . . 6 $/\$
27	24, 25, 26	sylancl 411	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 13577	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 7950	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 13578	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 7867	. . . . . . . . 9
34	24	recnd 7948	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8127	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 412	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2206	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 7948	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 7938	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4011	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 7935	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 8790	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1237	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 146	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9649	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 412	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 11659	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4011	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 13490	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 409	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 166	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 7948	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulid1d 7937	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 11612	. . . . . . . . 9
57		reeflog 13578	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4017	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4025	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 13491	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 412	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 166	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8374	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 412	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 8773	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1237	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4013	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8342	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 13585	. . . . 5 $/\$
73	15, 20, 72	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
74		1cnd 7936	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 7948	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8334	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 7948	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9659	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8731	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 7938	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 5871	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2203	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4020	. . 3 $/\$ $/\$ $/\$
84		relogcl 13577	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9685	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 7950	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8473	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 166	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9705	. 2 $/\$ $/\$ $/\$
91	89, 90	mpbird 166	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 973

wceq 1348

wcel 2141 class class class wbr 3989

cfv 5198 (class class class)co 5853

cc 7772

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

clt 7954

cle 7955

cmin 8090

cdiv 8589

crp 9610

ce 11605

ceu 11606

clog 13571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 ax-pre-suploc 7895 ax-addf 7896 ax-mulf 7897

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-disj 3967 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-of 6061 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-map 6628 df-pm 6629 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-xneg 9729 df-xadd 9730 df-ioo 9849 df-ico 9851 df-icc 9852 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-fac 10660 df-bc 10682 df-ihash 10710 df-shft 10779 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 df-ef 11611 df-e 11612 df-rest 12581 df-topgen 12600 df-psmet 12781 df-xmet 12782 df-met 12783 df-bl 12784 df-mopn 12785 df-top 12790 df-topon 12803 df-bases 12835 df-ntr 12890 df-cn 12982 df-cnp 12983 df-tx 13047 df-cncf 13352 df-limced 13419 df-dvap 13420 df-relog 13573

This theorem is referenced by: (None)

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