amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11828

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 9325	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 531	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8271	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8910	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11739	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8318	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10987	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 11021	. . . . . . 7
13	12	oveq1i 6068	. . . . . 6
14		resqrtth 11741	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 6074	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2279	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2267	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8671	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 11087	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8318	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8318	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 11037	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 11039	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 6076	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 11086	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9324	. . . . . . . . . . . 12
30		remulcl 8271	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8319	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8318	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8671	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8318	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 11086	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8318	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8638	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8318	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9498	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9409	. . . . . . . . . . 11
42	41	oveq1i 6068	. . . . . . . . . 10
43		2cnd 9327	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8318	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8313	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2281	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8318	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8639	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2278	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2271	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8319	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 11086	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8318	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 11086	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8318	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9331	. . . . . . . . . 10
57		remulcl 8271	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8318	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8494	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4142	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8826	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4136	. . 3 $/\$ $/\$ $/\$
67		remulcl 8271	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11743	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9344	. . . . . 6
72		mulge0 8910	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8742	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 592	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 11092	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9345	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 9173	. . 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 2205 class class class wbr 4114

cfv 5357 (class class class)co 6058

cc 8141

cr 8142

cc0 8143

caddc 8146

cmul 8148

clt 8324

cle 8325

cmin 8460

cdiv 8963

c2 9305

c4 9307

cexp 10924

csqrt 11706

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-rp 10005 df-seqfrec 10834 df-exp 10925 df-rsqrt 11708

This theorem is referenced by: (None)

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