amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 10922

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 8815	. . . . . 6
2		simpll 519	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 7772	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8405	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 10833	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 7818	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10386	. . . . . 6 $/\$
11	1, 9, 10	sylancr 411	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10419	. . . . . . 7
13	12	oveq1i 5792	. . . . . 6
14		resqrtth 10835	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5798	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	syl5eq 2185	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2173	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8167	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10482	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 7818	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 7818	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10434	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10436	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5800	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10481	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 8814	. . . . . . . . . . . 12
30		remulcl 7772	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 7819	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 7818	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8167	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 7818	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10481	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 7818	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8135	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 7818	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 8986	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 8898	. . . . . . . . . . 11
42	41	oveq1i 5792	. . . . . . . . . 10
43		2cnd 8817	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 7818	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 7813	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	syl5eqr 2187	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 7818	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8136	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2184	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2177	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 7819	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10481	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 7818	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10481	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 7818	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 8821	. . . . . . . . . 10
57		remulcl 7772	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 411	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 7818	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 7991	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1217	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 3964	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8321	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 146	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 3958	. . 3 $/\$ $/\$ $/\$
67		remulcl 7772	. . . . 5 $/\$
68	29, 8, 67	sylancr 411	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 10837	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 409	. . . . 5 $/\$ $/\$ $/\$
71		0le2 8834	. . . . . 6
72		mulge0 8405	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 433	. . . . 5 $/\$
74	8, 70, 73	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
75		addge0 8237	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 578	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10487	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 166	. 2 $/\$ $/\$ $/\$
79		2pos 8835	. . . . 5
80	29, 79	pm3.2i 270	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8664	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1217	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 146	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1332

wcel 1481 class class class wbr 3937

cfv 5131 (class class class)co 5782

cc 7642

cr 7643

cc0 7644

caddc 7647

cmul 7649

clt 7824

cle 7825

cmin 7957

cdiv 8456

c2 8795

c4 8797

cexp 10323

csqrt 10800

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-seqfrec 10250 df-exp 10324 df-rsqrt 10802

This theorem is referenced by: (None)

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