amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 10883

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 8784	. . . . . 6
2		simpll 518	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 520	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 7741	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 408	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8374	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 10794	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 408	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 7787	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10348	. . . . . 6 $/\$
11	1, 9, 10	sylancr 410	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10381	. . . . . . 7
13	12	oveq1i 5777	. . . . . 6
14		resqrtth 10796	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 408	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5783	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	syl5eq 2182	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2170	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8136	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10444	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 7787	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 7787	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10396	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 408	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10398	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 408	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5785	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10443	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 8783	. . . . . . . . . . . 12
30		remulcl 7741	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 410	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 7788	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 7787	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8136	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 7787	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10443	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 7787	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8104	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 7787	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 8955	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 8867	. . . . . . . . . . 11
42	41	oveq1i 5777	. . . . . . . . . 10
43		2cnd 8786	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 7787	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 7782	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	syl5eqr 2184	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 7787	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8105	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2181	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2174	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 7788	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10443	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 7787	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10443	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 7787	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 8790	. . . . . . . . . 10
57		remulcl 7741	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 410	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 7787	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 7960	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1216	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 3951	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8290	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 146	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 3945	. . 3 $/\$ $/\$ $/\$
67		remulcl 7741	. . . . 5 $/\$
68	29, 8, 67	sylancr 410	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 10798	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 408	. . . . 5 $/\$ $/\$ $/\$
71		0le2 8803	. . . . . 6
72		mulge0 8374	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 432	. . . . 5 $/\$
74	8, 70, 73	syl2anc 408	. . . 4 $/\$ $/\$ $/\$
75		addge0 8206	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 577	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10449	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 166	. 2 $/\$ $/\$ $/\$
79		2pos 8804	. . . . 5
80	29, 79	pm3.2i 270	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8633	. . 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 3924

cfv 5118 (class class class)co 5767

cc 7611

cr 7612

cc0 7613

caddc 7616

cmul 7618

clt 7793

cle 7794

cmin 7926

cdiv 8425

c2 8764

c4 8766

cexp 10285

csqrt 10761

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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733

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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-rp 9435 df-seqfrec 10212 df-exp 10286 df-rsqrt 10763

This theorem is referenced by: (None)

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