amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11262

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 9053	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8000	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8638	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11173	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8048	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10672	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10706	. . . . . . 7
13	12	oveq1i 5928	. . . . . 6
14		resqrtth 11175	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5934	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2238	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2226	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8400	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10771	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8048	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8048	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10722	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10724	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5936	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10770	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9052	. . . . . . . . . . . 12
30		remulcl 8000	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8049	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8048	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8400	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8048	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10770	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8048	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8368	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8048	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9225	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9136	. . . . . . . . . . 11
42	41	oveq1i 5928	. . . . . . . . . 10
43		2cnd 9055	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8048	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8043	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2240	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8048	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8369	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2237	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2230	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8049	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10770	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8048	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10770	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8048	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9059	. . . . . . . . . 10
57		remulcl 8000	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8048	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8224	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4057	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8554	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4051	. . 3 $/\$ $/\$ $/\$
67		remulcl 8000	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11177	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9072	. . . . . 6
72		mulge0 8638	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8470	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 588	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10776	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9073	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8901	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1249	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2164 class class class wbr 4029

cfv 5254 (class class class)co 5918

cc 7870

cr 7871

cc0 7872

caddc 7875

cmul 7877

clt 8054

cle 8055

cmin 8190

cdiv 8691

c2 9033

c4 9035

cexp 10609

csqrt 11140

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-rp 9720 df-seqfrec 10519 df-exp 10610 df-rsqrt 11142

This theorem is referenced by: (None)

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