amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11862

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 9354	. . . . . 6
2		simpll 531	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 535	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8297	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8937	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11773	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8344	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 11016	. . . . . 6 $/\$
11	1, 9, 10	sylancr 418	. . . . 5 $/\$ $/\$ $/\$
12		sq2 11050	. . . . . . 7
13	12	oveq1i 6085	. . . . . 6
14		resqrtth 11775	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 6091	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2283	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2271	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8698	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 11116	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8344	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8344	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 11066	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 11068	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 6093	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 11115	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9353	. . . . . . . . . . . 12
30		remulcl 8297	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 418	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8345	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8344	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8698	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8344	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 11115	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8344	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8665	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8344	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9527	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9438	. . . . . . . . . . 11
42	41	oveq1i 6085	. . . . . . . . . 10
43		2cnd 9356	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8344	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8339	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2285	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8344	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8666	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2282	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2275	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8345	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 11115	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8344	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 11115	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8344	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9360	. . . . . . . . . 10
57		remulcl 8297	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 418	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8344	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8521	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4153	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8853	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4147	. . 3 $/\$ $/\$ $/\$
67		remulcl 8297	. . . . 5 $/\$
68	29, 8, 67	sylancr 418	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11777	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 415	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9373	. . . . . 6
72		mulge0 8937	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 440	. . . . 5 $/\$
74	8, 70, 73	syl2anc 415	. . . 4 $/\$ $/\$ $/\$
75		addge0 8769	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 596	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 11121	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9374	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 9202	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1278	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1402

wcel 2209 class class class wbr 4125

cfv 5372 (class class class)co 6075

cc 8167

cr 8168

cc0 8169

caddc 8172

cmul 8174

clt 8350

cle 8351

cmin 8487

cdiv 8992

c2 9334

c4 9336

cexp 10953

csqrt 11740

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 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 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem 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 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-rp 10034 df-seqfrec 10863 df-exp 10954 df-rsqrt 11742

This theorem is referenced by: (None)

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