logdivlt - Intuitionistic Logic Explorer

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Theorem logdivlt 16136

Description: The

function is strictly decreasing on the reals greater than

. (Contributed by Mario Carneiro, 14-Mar-2014.)

**Assertion**
Ref	Expression
logdivlt	$/\$ $/\$ $/\$

**Proof of Theorem logdivlt**
Step	Hyp	Ref	Expression
1		logdivlti 16123	. . . . . 6 $/\$ $/\$ $/\$
2	1	ex 115	. . . . 5 $/\$ $/\$
3	2	3expa 1234	. . . 4 $/\$ $/\$
4	3	an32s 574	. . 3 $/\$ $/\$
5	4	adantrr 483	. 2 $/\$ $/\$ $/\$
6		simpll 531	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
7	6	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0red 8334	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		ere 12463	. . . . . . . . . . . . . . 15
10	9	a1i 9	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
11		epos 12574	. . . . . . . . . . . . . . 15
12	11	a1i 9	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
13		simpllr 540	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	8, 10, 7, 12, 13	ltletrd 8759	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 14	elrpd 10112	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16	15	relogcld 16124	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
17	16	recnd 8361	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	17	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ #
19	18	mulridd 8350	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ #
20		simpr 110	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ # #
21		simprl 535	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
22		0red 8334	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
23	9	a1i 9	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
24	11	a1i 9	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
25		simprr 537	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
26	22, 23, 21, 24, 25	ltletrd 8759	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
27	21, 26	elrpd 10112	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
28	27	relogcld 16124	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
29	28	recnd 8361	. . . . . . . . . . 11 $/\$ $/\$ $/\$
30	29	ad2antrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ #
31		1cnd 8349	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ #
32		simplr 533	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
33	22, 23, 6, 24, 32	ltletrd 8759	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
34	6, 33	elrpd 10112	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
35	34	relogcld 16124	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
36		1red 8348	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
37		2re 9384	. . . . . . . . . . . . . . 15
38	37	a1i 9	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
39		1lt2 9486	. . . . . . . . . . . . . . 15
40	39	a1i 9	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
41		egt2lt3 12573	. . . . . . . . . . . . . . . . 17 $/\$
42	41	simpli 111	. . . . . . . . . . . . . . . 16
43	42	a1i 9	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
44	38, 23, 6, 43, 32	ltletrd 8759	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
45	36, 38, 6, 40, 44	lttrd 8460	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
46		loggt0b 16133	. . . . . . . . . . . . . 14
47	34, 46	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
48	45, 47	mpbird 167	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
49	35, 48	gt0ap0d 8966	. . . . . . . . . . 11 $/\$ $/\$ $/\$ #
50	49	ad2antrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ # #
51	30, 18, 31, 50	apdivmuld 9152	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ # # #
52	20, 51	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ # #
53	19, 52	eqbrtrrd 4155	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ # #
54	15	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ #
55	21	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
56		simplrr 542	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
57	8, 10, 55, 12, 56	ltletrd 8759	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
58	55, 57	elrpd 10112	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
59	58	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ #
60		reaplog 16109	. . . . . . . 8 $/\$ # #
61	54, 59, 60	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ # # #
62	53, 61	mpbird 167	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ # #
63	7	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ #
64	63	recnd 8361	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ #
65	64	mulridd 8350	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ #
66		simpr 110	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ # #
67	55	recnd 8361	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
68	67	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ #
69		1cnd 8349	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ #
70	6, 33	gt0ap0d 8966	. . . . . . . . . 10 $/\$ $/\$ $/\$ #
71	70	ad2antrr 492	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ # #
72	68, 64, 69, 71	apdivmuld 9152	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ # # #
73	66, 72	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ # #
74	65, 73	eqbrtrrd 4155	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ # #
75	58	relogcld 16124	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
76	48	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
77	16, 76	elrpd 10112	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
78	75, 77	rerpdivcld 10147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
79	55, 15	rerpdivcld 10147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
80		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81	75, 15, 16, 58	lt2mul2divd 10184	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
82	80, 81	mpbird 167	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
83	17, 67	mulcomd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
84	82, 83	breqtrd 4157	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
85	75, 15, 55, 77	lt2mul2divd 10184	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
86	84, 85	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
87	78, 79, 86	ltapd 8975	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ #
88	78	recnd 8361	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
89	79	recnd 8361	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
90		1cnd 8349	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
91		apcotr 8944	. . . . . . . 8 $/\$ $/\$ # # $\/$ #
92	88, 89, 90, 91	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ # # $\/$ #
93	87, 92	mpd 13	. . . . . 6 $/\$ $/\$ $/\$ $/\$ # $\/$ #
94	62, 74, 93	mpjaodan 810	. . . . 5 $/\$ $/\$ $/\$ $/\$ #
95	7	recnd 8361	. . . . . 6 $/\$ $/\$ $/\$ $/\$
96		apsym 8943	. . . . . 6 $/\$ # #
97	95, 67, 96	syl2anc 415	. . . . 5 $/\$ $/\$ $/\$ $/\$ # #
98	94, 97	mpbid 147	. . . 4 $/\$ $/\$ $/\$ $/\$ #
99	75, 58	rerpdivcld 10147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
100	16, 15	rerpdivcld 10147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
101	99, 100, 80	ltnsymd 8454	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
102		logdivlti 16123	. . . . . . . . . 10 $/\$ $/\$ $/\$
103	102	ex 115	. . . . . . . . 9 $/\$ $/\$
104	103	con3d 640	. . . . . . . 8 $/\$ $/\$
105	55, 7, 56, 104	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
106	101, 105	mpd 13	. . . . . 6 $/\$ $/\$ $/\$ $/\$
107	6, 21	lenltd 8452	. . . . . . 7 $/\$ $/\$ $/\$
108	107	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$
109	106, 108	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$
110	7, 55, 109	leltapd 8976	. . . 4 $/\$ $/\$ $/\$ $/\$ #
111	98, 110	mpbird 167	. . 3 $/\$ $/\$ $/\$ $/\$
112	111	ex 115	. 2 $/\$ $/\$ $/\$
113	5, 112	impbid 129	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 $/\$ w3a 1009

wcel 2209 class class class wbr 4131

cfv 5380 (class class class)co 6088

cc 8184

cr 8185

cc0 8186

c1 8187

cmul 8191

clt 8367

cle 8368 # cap 8918

cdiv 9011

c2 9365

c3 9366

crp 10072

ceu 12436

clog 16097

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306 ax-pre-suploc 8307 ax-addf 8308 ax-mulf 8309

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-map 6928 df-pm 6929 df-en 7028 df-dom 7029 df-fin 7030 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-q 10037 df-rp 10073 df-xneg 10192 df-xadd 10193 df-ioo 10312 df-ico 10314 df-icc 10315 df-fz 10430 df-fzo 10568 df-seqfrec 10907 df-exp 10998 df-fac 11187 df-bc 11209 df-ihash 11238 df-shft 11603 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788 df-clim 12071 df-sumdc 12146 df-ef 12441 df-e 12442 df-rest 13655 df-topgen 13674 df-psmet 14990 df-xmet 14991 df-met 14992 df-bl 14993 df-mopn 14994 df-top 15236 df-topon 15249 df-bases 15281 df-ntr 15334 df-cn 15426 df-cnp 15427 df-tx 15491 df-cncf 15809 df-limced 15894 df-dvap 15895 df-relog 16099

This theorem is used by: logdivle 16137 bposlem7 16329

W3C validator