amgm2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > amgm2		Unicode version

Theorem amgm2 11300

Description: Arithmetic-geometric mean inequality for

. (Contributed by Mario Carneiro, 2-Jul-2014.)

**Assertion**
Ref	Expression
amgm2	$/\$ $/\$ $/\$

**Proof of Theorem amgm2**
Step	Hyp	Ref	Expression
1		2cn 9078	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8024	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8663	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11211	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8072	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10710	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10744	. . . . . . 7
13	12	oveq1i 5935	. . . . . 6
14		resqrtth 11213	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5941	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2241	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2229	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8424	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10809	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8072	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8072	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10760	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10762	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5943	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10808	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9077	. . . . . . . . . . . 12
30		remulcl 8024	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8073	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8072	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8424	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8072	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10808	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8072	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8392	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8072	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9251	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9162	. . . . . . . . . . 11
42	41	oveq1i 5935	. . . . . . . . . 10
43		2cnd 9080	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8072	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8067	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2243	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8072	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8393	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2240	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2233	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8073	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10808	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8072	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10808	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8072	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9084	. . . . . . . . . 10
57		remulcl 8024	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8072	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8248	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4062	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8579	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4056	. . 3 $/\$ $/\$ $/\$
67		remulcl 8024	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11215	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9097	. . . . . 6
72		mulge0 8663	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8495	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 588	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10814	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9098	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8926	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1249	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2167 class class class wbr 4034

cfv 5259 (class class class)co 5925

cc 7894

cr 7895

cc0 7896

caddc 7899

cmul 7901

clt 8078

cle 8079

cmin 8214

cdiv 8716

c2 9058

c4 9060

cexp 10647

csqrt 11178

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 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

This theorem depends on definitions: df-bi 117 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-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-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-rp 9746 df-seqfrec 10557 df-exp 10648 df-rsqrt 11180

This theorem is referenced by: (None)

W3C validator