logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 14195

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 11673	. . . . . . . . . . 11
5		simpl1 1000	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8044	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1342	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 11783	. . . . . . . . . 10
10		0re 7956	. . . . . . . . . . 11
11		lttr 8029	. . . . . . . . . . 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 9691	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8045	. . . . . . . . . . 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 9691	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9712	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 14176	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9726	. . . . . 6 $/\$ $/\$ $/\$
25		1re 7955	. . . . . 6
26		resubcl 8219	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 14176	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 7986	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 14177	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 7903	. . . . . . . . 9
34	24	recnd 7984	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8163	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2213	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 7984	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 7974	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4025	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 7971	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 8829	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1242	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9690	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 11699	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4025	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 14089	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 7984	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 7973	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 11652	. . . . . . . . 9
57		reeflog 14177	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4031	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4039	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 14090	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8410	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 8812	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1242	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4027	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8378	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 14184	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 7972	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 7984	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8370	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 7984	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9700	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8769	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 7974	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 5892	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2210	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4034	. . 3 $/\$ $/\$ $/\$
84		relogcl 14176	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9726	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 7986	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8509	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9746	. 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 4003

cfv 5216 (class class class)co 5874

cc 7808

cr 7809

cc0 7810

c1 7811

caddc 7813

cmul 7815

clt 7990

cle 7991

cmin 8126

cdiv 8627

crp 9651

ce 11645

ceu 11646

clog 14170

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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 ax-pre-suploc 7931 ax-addf 7932 ax-mulf 7933

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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-disj 3981 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-of 6082 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-map 6649 df-pm 6650 df-en 6740 df-dom 6741 df-fin 6742 df-sup 6982 df-inf 6983 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-xneg 9770 df-xadd 9771 df-ioo 9890 df-ico 9892 df-icc 9893 df-fz 10007 df-fzo 10140 df-seqfrec 10443 df-exp 10517 df-fac 10701 df-bc 10723 df-ihash 10751 df-shft 10819 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-clim 11282 df-sumdc 11357 df-ef 11651 df-e 11652 df-rest 12680 df-topgen 12699 df-psmet 13338 df-xmet 13339 df-met 13340 df-bl 13341 df-mopn 13342 df-top 13389 df-topon 13402 df-bases 13434 df-ntr 13489 df-cn 13581 df-cnp 13582 df-tx 13646 df-cncf 13951 df-limced 14018 df-dvap 14019 df-relog 14172

This theorem is referenced by: (None)

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