amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11644

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 9192	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8138	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8777	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11555	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8186	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10835	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10869	. . . . . . 7
13	12	oveq1i 6017	. . . . . 6
14		resqrtth 11557	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 6023	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2274	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2262	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8538	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10934	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8186	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8186	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10885	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10887	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 6025	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10933	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9191	. . . . . . . . . . . 12
30		remulcl 8138	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8187	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8186	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8538	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8186	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10933	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8186	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8506	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8186	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9365	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9276	. . . . . . . . . . 11
42	41	oveq1i 6017	. . . . . . . . . 10
43		2cnd 9194	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8186	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8181	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2276	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8186	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8507	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2273	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2266	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8187	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10933	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8186	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10933	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8186	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9198	. . . . . . . . . 10
57		remulcl 8138	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8186	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8362	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1271	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4111	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8693	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4105	. . 3 $/\$ $/\$ $/\$
67		remulcl 8138	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11559	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9211	. . . . . 6
72		mulge0 8777	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8609	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 590	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10939	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9212	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 9040	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1271	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1395

wcel 2200 class class class wbr 4083

cfv 5318 (class class class)co 6007

cc 8008

cr 8009

cc0 8010

caddc 8013

cmul 8015

clt 8192

cle 8193

cmin 8328

cdiv 8830

c2 9172

c4 9174

cexp 10772

csqrt 11522

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-rp 9862 df-seqfrec 10682 df-exp 10773 df-rsqrt 11524

This theorem is referenced by: (None)

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