amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11803

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 9308	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 531	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8255	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8893	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11714	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8302	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10963	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10997	. . . . . . 7
13	12	oveq1i 6060	. . . . . 6
14		resqrtth 11716	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 6066	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2277	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2265	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8654	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 11062	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8302	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8302	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 11013	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 11015	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 6068	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 11061	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9307	. . . . . . . . . . . 12
30		remulcl 8255	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8303	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8302	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8654	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8302	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 11061	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8302	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8622	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8302	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9481	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9392	. . . . . . . . . . 11
42	41	oveq1i 6060	. . . . . . . . . 10
43		2cnd 9310	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8302	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8297	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2279	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8302	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8623	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2276	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2269	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8303	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 11061	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8302	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 11061	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8302	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9314	. . . . . . . . . 10
57		remulcl 8255	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8302	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8478	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4137	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8809	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4131	. . 3 $/\$ $/\$ $/\$
67		remulcl 8255	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11718	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9327	. . . . . 6
72		mulge0 8893	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8725	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 592	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 11067	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9328	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 9156	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1274	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2203 class class class wbr 4109

cfv 5352 (class class class)co 6050

cc 8125

cr 8126

cc0 8127

caddc 8130

cmul 8132

clt 8308

cle 8309

cmin 8444

cdiv 8946

c2 9288

c4 9290

cexp 10900

csqrt 11681

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-rp 9987 df-seqfrec 10810 df-exp 10901 df-rsqrt 11683

This theorem is referenced by: (None)

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