logdivlti - Intuitionistic Logic Explorer

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Theorem logdivlti 15225

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 1003	. . . . . . . 8 $/\$ $/\$ $/\$
2		simpl3 1004	. . . . . . . . . 10 $/\$ $/\$ $/\$
3		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$
4		ere 11854	. . . . . . . . . . 11
5		simpl1 1002	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		lelttr 8134	. . . . . . . . . . 11 $/\$ $/\$ $/\$
7	4, 5, 1, 6	mp3an2i 1353	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
8	2, 3, 7	mp2and 433	. . . . . . . . 9 $/\$ $/\$ $/\$
9		epos 11965	. . . . . . . . . 10
10		0re 8045	. . . . . . . . . . 11
11		lttr 8119	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	10, 4, 1, 11	mp3an12i 1352	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	9, 12	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
14	8, 13	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
15	1, 14	elrpd 9787	. . . . . . 7 $/\$ $/\$ $/\$
16		ltletr 8135	. . . . . . . . . . 11 $/\$ $/\$ $/\$
17	10, 4, 5, 16	mp3an12i 1352	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	9, 17	mpani 430	. . . . . . . . 9 $/\$ $/\$ $/\$
19	2, 18	mpd 13	. . . . . . . 8 $/\$ $/\$ $/\$
20	5, 19	elrpd 9787	. . . . . . 7 $/\$ $/\$ $/\$
21	15, 20	rpdivcld 9808	. . . . . 6 $/\$ $/\$ $/\$
22		relogcl 15206	. . . . . 6
23	21, 22	syl 14	. . . . 5 $/\$ $/\$ $/\$
24	1, 20	rerpdivcld 9822	. . . . . 6 $/\$ $/\$ $/\$
25		1re 8044	. . . . . 6
26		resubcl 8309	. . . . . 6 $/\$
27	24, 25, 26	sylancl 413	. . . . 5 $/\$ $/\$ $/\$
28		relogcl 15206	. . . . . . 7
29	20, 28	syl 14	. . . . . 6 $/\$ $/\$ $/\$
30	27, 29	remulcld 8076	. . . . 5 $/\$ $/\$ $/\$
31		reeflog 15207	. . . . . . . . 9
32	21, 31	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
33		ax-1cn 7991	. . . . . . . . 9
34	24	recnd 8074	. . . . . . . . 9 $/\$ $/\$ $/\$
35		pncan3 8253	. . . . . . . . 9 $/\$
36	33, 34, 35	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
37	32, 36	eqtr4d 2232	. . . . . . 7 $/\$ $/\$ $/\$
38	5	recnd 8074	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
39	38	mulid2d 8064	. . . . . . . . . . 11 $/\$ $/\$ $/\$
40	39, 3	eqbrtrd 4056	. . . . . . . . . 10 $/\$ $/\$ $/\$
41		1red 8060	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		ltmuldiv 8920	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43	41, 1, 5, 19, 42	syl112anc 1253	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	40, 43	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$
45		difrp 9786	. . . . . . . . . 10 $/\$
46	25, 24, 45	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
47	44, 46	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
48		efgt1p 11880	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
50	37, 49	eqbrtrd 4056	. . . . . 6 $/\$ $/\$ $/\$
51		eflt 15119	. . . . . . 7 $/\$
52	23, 27, 51	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
53	50, 52	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
54	27	recnd 8074	. . . . . . 7 $/\$ $/\$ $/\$
55	54	mulridd 8062	. . . . . 6 $/\$ $/\$ $/\$
56		df-e 11833	. . . . . . . . 9
57		reeflog 15207	. . . . . . . . . . 11
58	20, 57	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
59	2, 58	breqtrrd 4062	. . . . . . . . 9 $/\$ $/\$ $/\$
60	56, 59	eqbrtrrid 4070	. . . . . . . 8 $/\$ $/\$ $/\$
61		efle 15120	. . . . . . . . 9 $/\$
62	25, 29, 61	sylancr 414	. . . . . . . 8 $/\$ $/\$ $/\$
63	60, 62	mpbird 167	. . . . . . 7 $/\$ $/\$ $/\$
64		posdif 8501	. . . . . . . . . 10 $/\$
65	25, 24, 64	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
66	44, 65	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$
67		lemul2 8903	. . . . . . . 8 $/\$ $/\$ $/\$
68	41, 29, 27, 66, 67	syl112anc 1253	. . . . . . 7 $/\$ $/\$ $/\$
69	63, 68	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
70	55, 69	eqbrtrrd 4058	. . . . 5 $/\$ $/\$ $/\$
71	23, 27, 30, 53, 70	ltletrd 8469	. . . 4 $/\$ $/\$ $/\$
72		relogdiv 15214	. . . . 5 $/\$
73	15, 20, 72	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
74		1cnd 8061	. . . . . 6 $/\$ $/\$ $/\$
75	29	recnd 8074	. . . . . 6 $/\$ $/\$ $/\$
76	34, 74, 75	subdird 8460	. . . . 5 $/\$ $/\$ $/\$
77	1	recnd 8074	. . . . . . 7 $/\$ $/\$ $/\$
78	20	rpap0d 9796	. . . . . . 7 $/\$ $/\$ $/\$ #
79	77, 38, 75, 78	div32apd 8860	. . . . . 6 $/\$ $/\$ $/\$
80	75	mulid2d 8064	. . . . . 6 $/\$ $/\$ $/\$
81	79, 80	oveq12d 5943	. . . . 5 $/\$ $/\$ $/\$
82	76, 81	eqtrd 2229	. . . 4 $/\$ $/\$ $/\$
83	71, 73, 82	3brtr3d 4065	. . 3 $/\$ $/\$ $/\$
84		relogcl 15206	. . . . 5
85	15, 84	syl 14	. . . 4 $/\$ $/\$ $/\$
86	29, 20	rerpdivcld 9822	. . . . 5 $/\$ $/\$ $/\$
87	1, 86	remulcld 8076	. . . 4 $/\$ $/\$ $/\$
88	85, 87, 29	ltsub1d 8600	. . 3 $/\$ $/\$ $/\$
89	83, 88	mpbird 167	. 2 $/\$ $/\$ $/\$
90	85, 86, 15	ltdivmuld 9842	. 2 $/\$ $/\$ $/\$
91	89, 90	mpbird 167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

wcel 2167 class class class wbr 4034

cfv 5259 (class class class)co 5925

cc 7896

cr 7897

cc0 7898

c1 7899

caddc 7901

cmul 7903

clt 8080

cle 8081

cmin 8216

cdiv 8718

crp 9747

ce 11826

ceu 11827

clog 15200

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017 ax-caucvg 8018 ax-pre-suploc 8019 ax-addf 8020 ax-mulf 8021

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-disj 4012 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-map 6718 df-pm 6719 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-n0 9269 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-xneg 9866 df-xadd 9867 df-ioo 9986 df-ico 9988 df-icc 9989 df-fz 10103 df-fzo 10237 df-seqfrec 10559 df-exp 10650 df-fac 10837 df-bc 10859 df-ihash 10887 df-shft 10999 df-cj 11026 df-re 11027 df-im 11028 df-rsqrt 11182 df-abs 11183 df-clim 11463 df-sumdc 11538 df-ef 11832 df-e 11833 df-rest 12945 df-topgen 12964 df-psmet 14177 df-xmet 14178 df-met 14179 df-bl 14180 df-mopn 14181 df-top 14342 df-topon 14355 df-bases 14387 df-ntr 14440 df-cn 14532 df-cnp 14533 df-tx 14597 df-cncf 14915 df-limced 15000 df-dvap 15001 df-relog 15202

This theorem is referenced by: (None)

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