amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11126

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 8989	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 7938	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8575	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11037	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 7985	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10581	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10615	. . . . . . 7
13	12	oveq1i 5884	. . . . . 6
14		resqrtth 11039	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5890	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2222	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2210	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8337	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10680	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 7985	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 7985	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10631	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10633	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5892	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10679	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 8988	. . . . . . . . . . . 12
30		remulcl 7938	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 7986	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 7985	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8337	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 7985	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10679	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 7985	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8305	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 7985	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9160	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9072	. . . . . . . . . . 11
42	41	oveq1i 5884	. . . . . . . . . 10
43		2cnd 8991	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 7985	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 7980	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2224	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 7985	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8306	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2221	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2214	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 7986	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10679	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 7985	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10679	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 7985	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 8995	. . . . . . . . . 10
57		remulcl 7938	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 7985	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8161	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1238	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4031	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8491	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4025	. . 3 $/\$ $/\$ $/\$
67		remulcl 7938	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11041	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9008	. . . . . 6
72		mulge0 8575	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8407	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 588	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10685	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9009	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8838	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1238	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148 class class class wbr 4003

cfv 5216 (class class class)co 5874

cc 7808

cr 7809

cc0 7810

caddc 7813

cmul 7815

clt 7991

cle 7992

cmin 8127

cdiv 8628

c2 8969

c4 8971

cexp 10518

csqrt 11004

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-n0 9176 df-z 9253 df-uz 9528 df-rp 9653 df-seqfrec 10445 df-exp 10519 df-rsqrt 11006

This theorem is referenced by: (None)

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