amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11624

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 9177	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8123	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8762	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11535	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8171	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10818	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10852	. . . . . . 7
13	12	oveq1i 6010	. . . . . 6
14		resqrtth 11537	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 6016	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2274	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2262	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8523	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10917	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8171	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8171	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10868	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10870	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 6018	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10916	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9176	. . . . . . . . . . . 12
30		remulcl 8123	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8172	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8171	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8523	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8171	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10916	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8171	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8491	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8171	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9350	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9261	. . . . . . . . . . 11
42	41	oveq1i 6010	. . . . . . . . . 10
43		2cnd 9179	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8171	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8166	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2276	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8171	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8492	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2273	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2266	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8172	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10916	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8171	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10916	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8171	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9183	. . . . . . . . . 10
57		remulcl 8123	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8171	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8347	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1271	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4110	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8678	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4104	. . 3 $/\$ $/\$ $/\$
67		remulcl 8123	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11539	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9196	. . . . . 6
72		mulge0 8762	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8594	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 590	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10922	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9197	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 9025	. . 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 4082

cfv 5317 (class class class)co 6000

cc 7993

cr 7994

cc0 7995

caddc 7998

cmul 8000

clt 8177

cle 8178

cmin 8313

cdiv 8815

c2 9157

c4 9159

cexp 10755

csqrt 11502

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 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

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 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-rp 9846 df-seqfrec 10665 df-exp 10756 df-rsqrt 11504

This theorem is referenced by: (None)

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