amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11669

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 9204	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8150	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8789	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11580	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8198	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10853	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10887	. . . . . . 7
13	12	oveq1i 6023	. . . . . 6
14		resqrtth 11582	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 6029	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2274	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2262	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8550	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10952	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8198	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8198	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10903	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10905	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 6031	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10951	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9203	. . . . . . . . . . . 12
30		remulcl 8150	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8199	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8198	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8550	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8198	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10951	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8198	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8518	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8198	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9377	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9288	. . . . . . . . . . 11
42	41	oveq1i 6023	. . . . . . . . . 10
43		2cnd 9206	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8198	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8193	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2276	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8198	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8519	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2273	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2266	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8199	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10951	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8198	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10951	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8198	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9210	. . . . . . . . . 10
57		remulcl 8150	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8198	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8374	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1271	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4114	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8705	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4108	. . 3 $/\$ $/\$ $/\$
67		remulcl 8150	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11584	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9223	. . . . . 6
72		mulge0 8789	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8621	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 590	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10957	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9224	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 9052	. . 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 4086

cfv 5324 (class class class)co 6013

cc 8020

cr 8021

cc0 8022

caddc 8025

cmul 8027

clt 8204

cle 8205

cmin 8340

cdiv 8842

c2 9184

c4 9186

cexp 10790

csqrt 11547

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 ax-arch 8141 ax-caucvg 8142

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-n0 9393 df-z 9470 df-uz 9746 df-rp 9879 df-seqfrec 10700 df-exp 10791 df-rsqrt 11549

This theorem is referenced by: (None)

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