amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11884

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 9375	. . . . . 6
2		simpll 531	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 535	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8307	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8947	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11795	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8354	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 11038	. . . . . 6 $/\$
11	1, 9, 10	sylancr 418	. . . . 5 $/\$ $/\$ $/\$
12		sq2 11072	. . . . . . 7
13	12	oveq1i 6095	. . . . . 6
14		resqrtth 11797	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 6101	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2283	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2271	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8708	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 11138	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 11088	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 11090	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 6103	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 11137	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9374	. . . . . . . . . . . 12
30		remulcl 8307	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 418	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8355	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8354	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8708	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8354	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 11137	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8354	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8675	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9548	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9459	. . . . . . . . . . 11
42	41	oveq1i 6095	. . . . . . . . . 10
43		2cnd 9377	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8354	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8349	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2285	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8676	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2282	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2275	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8355	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 11137	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8354	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 11137	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8354	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9381	. . . . . . . . . 10
57		remulcl 8307	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 418	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8354	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8531	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4158	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8863	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4152	. . 3 $/\$ $/\$ $/\$
67		remulcl 8307	. . . . 5 $/\$
68	29, 8, 67	sylancr 418	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11799	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 415	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9394	. . . . . 6
72		mulge0 8947	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 440	. . . . 5 $/\$
74	8, 70, 73	syl2anc 415	. . . 4 $/\$ $/\$ $/\$
75		addge0 8779	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 596	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 11143	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9395	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 9212	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1278	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105

wceq 1402

wcel 2209 class class class wbr 4130

cfv 5377 (class class class)co 6085

cc 8177

cr 8178

cc0 8179

caddc 8182

cmul 8184

clt 8360

cle 8361

cmin 8497

cdiv 9002

c2 9355

c4 9357

cexp 10975

csqrt 11762

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-rp 10055 df-seqfrec 10885 df-exp 10976 df-rsqrt 11764

This theorem is used by: (None)

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