amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 10890

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 8791	. . . . . 6
2		simpll 518	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 520	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 7748	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 408	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8381	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 10801	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 408	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 7794	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10355	. . . . . 6 $/\$
11	1, 9, 10	sylancr 410	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10388	. . . . . . 7
13	12	oveq1i 5784	. . . . . 6
14		resqrtth 10803	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 408	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5790	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	syl5eq 2184	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2172	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8143	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10451	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 7794	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 7794	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10403	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 408	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10405	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 408	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5792	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10450	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 8790	. . . . . . . . . . . 12
30		remulcl 7748	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 410	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 7795	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 7794	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8143	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 7794	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10450	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 7794	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8111	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 7794	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 8962	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 8874	. . . . . . . . . . 11
42	41	oveq1i 5784	. . . . . . . . . 10
43		2cnd 8793	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 7794	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 7789	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	syl5eqr 2186	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 7794	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8112	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2183	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2176	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 7795	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10450	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 7794	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10450	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 7794	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 8797	. . . . . . . . . 10
57		remulcl 7748	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 410	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 7794	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 7967	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1216	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 3956	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8297	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 146	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 3950	. . 3 $/\$ $/\$ $/\$
67		remulcl 7748	. . . . 5 $/\$
68	29, 8, 67	sylancr 410	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 10805	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 408	. . . . 5 $/\$ $/\$ $/\$
71		0le2 8810	. . . . . 6
72		mulge0 8381	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 432	. . . . 5 $/\$
74	8, 70, 73	syl2anc 408	. . . 4 $/\$ $/\$ $/\$
75		addge0 8213	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 577	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10456	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 166	. 2 $/\$ $/\$ $/\$
79		2pos 8811	. . . . 5
80	29, 79	pm3.2i 270	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8640	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1216	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 146	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1331

wcel 1480 class class class wbr 3929

cfv 5123 (class class class)co 5774

cc 7618

cr 7619

cc0 7620

caddc 7623

cmul 7625

clt 7800

cle 7801

cmin 7933

cdiv 8432

c2 8771

c4 8773

cexp 10292

csqrt 10768

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-seqfrec 10219 df-exp 10293 df-rsqrt 10770

This theorem is referenced by: (None)

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