amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11741

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 9256	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 531	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8203	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8841	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11652	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8250	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10909	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10943	. . . . . . 7
13	12	oveq1i 6038	. . . . . 6
14		resqrtth 11654	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 6044	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2276	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2264	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8602	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 11008	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8250	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8250	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10959	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10961	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 6046	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 11007	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9255	. . . . . . . . . . . 12
30		remulcl 8203	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8251	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8250	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8602	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8250	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 11007	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8250	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8570	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8250	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9429	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9340	. . . . . . . . . . 11
42	41	oveq1i 6038	. . . . . . . . . 10
43		2cnd 9258	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8250	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8245	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2278	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8250	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8571	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2275	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2268	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8251	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 11007	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8250	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 11007	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8250	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9262	. . . . . . . . . 10
57		remulcl 8203	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8250	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8426	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4121	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8757	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4115	. . 3 $/\$ $/\$ $/\$
67		remulcl 8203	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11656	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9275	. . . . . 6
72		mulge0 8841	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8673	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 592	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 11013	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9276	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 9104	. . 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 2202 class class class wbr 4093

cfv 5333 (class class class)co 6028

cc 8073

cr 8074

cc0 8075

caddc 8078

cmul 8080

clt 8256

cle 8257

cmin 8392

cdiv 8894

c2 9236

c4 9238

cexp 10846

csqrt 11619

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-n0 9445 df-z 9524 df-uz 9800 df-rp 9933 df-seqfrec 10756 df-exp 10847 df-rsqrt 11621

This theorem is referenced by: (None)

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