amgm2 - Intuitionistic Logic Explorer

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

Theorem amgm2 11130

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 8993	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 7942	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8579	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11041	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 7989	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10585	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10619	. . . . . . 7
13	12	oveq1i 5888	. . . . . 6
14		resqrtth 11043	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5894	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2222	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2210	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8341	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10684	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 7989	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 7989	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10635	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10637	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5896	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10683	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 8992	. . . . . . . . . . . 12
30		remulcl 7942	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 7990	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 7989	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8341	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 7989	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10683	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 7989	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8309	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 7989	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9164	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9076	. . . . . . . . . . 11
42	41	oveq1i 5888	. . . . . . . . . 10
43		2cnd 8995	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 7989	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 7984	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2224	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 7989	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8310	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2221	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2214	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 7990	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10683	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 7989	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10683	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 7989	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 8999	. . . . . . . . . 10
57		remulcl 7942	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 7989	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8165	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1238	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4033	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8495	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4027	. . 3 $/\$ $/\$ $/\$
67		remulcl 7942	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11045	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9012	. . . . . 6
72		mulge0 8579	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8411	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 588	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10689	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9013	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8842	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1238	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148 class class class wbr 4005

cfv 5218 (class class class)co 5878

cc 7812

cr 7813

cc0 7814

caddc 7817

cmul 7819

clt 7995

cle 7996

cmin 8131

cdiv 8632

c2 8973

c4 8975

cexp 10522

csqrt 11008

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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 ax-caucvg 7934

This theorem depends on definitions: df-bi 117 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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-frec 6395 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-rp 9657 df-seqfrec 10449 df-exp 10523 df-rsqrt 11010

This theorem is referenced by: (None)

W3C validator