logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 14305

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 11678	. . . . . . . . . . 11
5		simpl1 1000	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8046	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1342	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 11788	. . . . . . . . . 10
10		0re 7957	. . . . . . . . . . 11
11		lttr 8031	. . . . . . . . . . 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 9693	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8047	. . . . . . . . . . 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 9693	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9714	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 14286	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9728	. . . . . 6 $/\$ $/\$ $/\$
25		1re 7956	. . . . . 6
26		resubcl 8221	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 14286	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 7988	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 14287	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 7904	. . . . . . . . 9
34	24	recnd 7986	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8165	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2213	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 7986	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 7976	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4026	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 7972	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 8831	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1242	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9692	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 11704	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4026	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 14199	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 7986	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 7974	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 11657	. . . . . . . . 9
57		reeflog 14287	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4032	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4040	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 14200	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8412	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 8814	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1242	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4028	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8380	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 14294	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 7973	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 7986	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8372	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 7986	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9702	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8771	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 7976	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 5893	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2210	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4035	. . 3 $/\$ $/\$ $/\$
84		relogcl 14286	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9728	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 7988	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8511	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9748	. 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 4004

cfv 5217 (class class class)co 5875

cc 7809

cr 7810

cc0 7811

c1 7812

caddc 7814

cmul 7816

clt 7992

cle 7993

cmin 8128

cdiv 8629

crp 9653

ce 11650

ceu 11651

clog 14280

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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931 ax-pre-suploc 7932 ax-addf 7933 ax-mulf 7934

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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-disj 3982 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-of 6083 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-frec 6392 df-1o 6417 df-oadd 6421 df-er 6535 df-map 6650 df-pm 6651 df-en 6741 df-dom 6742 df-fin 6743 df-sup 6983 df-inf 6984 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-xneg 9772 df-xadd 9773 df-ioo 9892 df-ico 9894 df-icc 9895 df-fz 10009 df-fzo 10143 df-seqfrec 10446 df-exp 10520 df-fac 10706 df-bc 10728 df-ihash 10756 df-shft 10824 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-clim 11287 df-sumdc 11362 df-ef 11656 df-e 11657 df-rest 12690 df-topgen 12709 df-psmet 13450 df-xmet 13451 df-met 13452 df-bl 13453 df-mopn 13454 df-top 13501 df-topon 13514 df-bases 13546 df-ntr 13599 df-cn 13691 df-cnp 13692 df-tx 13756 df-cncf 14061 df-limced 14128 df-dvap 14129 df-relog 14282

This theorem is referenced by: (None)

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