amgm2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > amgm2		Structured version Visualization version Unicode version

Theorem amgm2 14109

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 11091	. . . . . 6
2		simpll 790	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 794	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 10021	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 10546	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 13994	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 10068	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 12926	. . . . . 6 $/\$
11	1, 9, 10	sylancr 695	. . . . 5 $/\$ $/\$ $/\$
12		sq2 12960	. . . . . . 7
13	12	oveq1i 6660	. . . . . 6
14	5	recnd 10068	. . . . . . . 8 $/\$ $/\$ $/\$
15		sqrtth 14104	. . . . . . . 8
16	14, 15	syl 17	. . . . . . 7 $/\$ $/\$ $/\$
17	16	oveq2d 6666	. . . . . 6 $/\$ $/\$ $/\$
18	13, 17	syl5eq 2668	. . . . 5 $/\$ $/\$ $/\$
19	11, 18	eqtrd 2656	. . . 4 $/\$ $/\$ $/\$
20	2, 3	resubcld 10458	. . . . . . 7 $/\$ $/\$ $/\$
21	20	sqge0d 13036	. . . . . 6 $/\$ $/\$ $/\$
22	2	recnd 10068	. . . . . . . . . 10 $/\$ $/\$ $/\$
23	3	recnd 10068	. . . . . . . . . 10 $/\$ $/\$ $/\$
24		binom2 12979	. . . . . . . . . 10 $/\$
25	22, 23, 24	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
26		binom2sub 12981	. . . . . . . . . 10 $/\$
27	22, 23, 26	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
28	25, 27	oveq12d 6668	. . . . . . . 8 $/\$ $/\$ $/\$
29	2	resqcld 13035	. . . . . . . . . . 11 $/\$ $/\$ $/\$
30		2re 11090	. . . . . . . . . . . 12
31		remulcl 10021	. . . . . . . . . . . 12 $/\$
32	30, 5, 31	sylancr 695	. . . . . . . . . . 11 $/\$ $/\$ $/\$
33	29, 32	readdcld 10069	. . . . . . . . . 10 $/\$ $/\$ $/\$
34	33	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$
35	29, 32	resubcld 10458	. . . . . . . . . 10 $/\$ $/\$ $/\$
36	35	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$
37	3	resqcld 13035	. . . . . . . . . 10 $/\$ $/\$ $/\$
38	37	recnd 10068	. . . . . . . . 9 $/\$ $/\$ $/\$
39	34, 36, 38	pnpcan2d 10430	. . . . . . . 8 $/\$ $/\$ $/\$
40	32	recnd 10068	. . . . . . . . . 10 $/\$ $/\$ $/\$
41	40	2timesd 11275	. . . . . . . . 9 $/\$ $/\$ $/\$
42		2t2e4 11177	. . . . . . . . . . 11
43	42	oveq1i 6660	. . . . . . . . . 10
44		2cnd 11093	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	44, 44, 14	mulassd 10063	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	43, 45	syl5eqr 2670	. . . . . . . . 9 $/\$ $/\$ $/\$
47	29	recnd 10068	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 40, 40	pnncand 10431	. . . . . . . . 9 $/\$ $/\$ $/\$
49	41, 46, 48	3eqtr4rd 2667	. . . . . . . 8 $/\$ $/\$ $/\$
50	28, 39, 49	3eqtrd 2660	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 10069	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 13035	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 10068	. . . . . . . 8 $/\$ $/\$ $/\$
54	20	resqcld 13035	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 10068	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 11097	. . . . . . . . . 10
57		remulcl 10021	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 695	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 10068	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 10286	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 222	. . . . . 6 $/\$ $/\$ $/\$
63	21, 62	breqtrrd 4681	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 10617	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 222	. . . 4 $/\$ $/\$ $/\$
66	19, 65	eqbrtrd 4675	. . 3 $/\$ $/\$ $/\$
67		remulcl 10021	. . . . 5 $/\$
68	30, 8, 67	sylancr 695	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 13998	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$
71		0le2 11111	. . . . . 6
72		mulge0 10546	. . . . . 6 $/\$ $/\$ $/\$
73	30, 71, 72	mpanl12 718	. . . . 5 $/\$
74	8, 70, 73	syl2anc 693	. . . 4 $/\$ $/\$ $/\$
75		addge0 10517	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 869	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 13044	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 247	. 2 $/\$ $/\$ $/\$
79		2pos 11112	. . . . 5
80	30, 79	pm3.2i 471	. . . 4 $/\$
81	80	a1i 11	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 10904	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1326	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 222	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cdiv 10684

c2 11070

c4 11072

cexp 12860

csqrt 13973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: (None)

W3C validator