amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11060

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 8928	. . . . . 6
2		simpll 519	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 7881	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8517	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 10971	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 7927	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10517	. . . . . 6 $/\$
11	1, 9, 10	sylancr 411	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10550	. . . . . . 7
13	12	oveq1i 5852	. . . . . 6
14		resqrtth 10973	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5858	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	syl5eq 2211	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2198	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8279	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10615	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 7927	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 7927	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10566	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10568	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5860	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10614	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 8927	. . . . . . . . . . . 12
30		remulcl 7881	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 7928	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 7927	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8279	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 7927	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10614	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 7927	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8247	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 7927	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9099	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9011	. . . . . . . . . . 11
42	41	oveq1i 5852	. . . . . . . . . 10
43		2cnd 8930	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 7927	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 7922	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2213	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 7927	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8248	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2209	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2202	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 7928	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10614	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 7927	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10614	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 7927	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 8934	. . . . . . . . . 10
57		remulcl 7881	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 411	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 7927	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8103	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1228	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4010	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8433	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 146	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4004	. . 3 $/\$ $/\$ $/\$
67		remulcl 7881	. . . . 5 $/\$
68	29, 8, 67	sylancr 411	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 10975	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 409	. . . . 5 $/\$ $/\$ $/\$
71		0le2 8947	. . . . . 6
72		mulge0 8517	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 433	. . . . 5 $/\$
74	8, 70, 73	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
75		addge0 8349	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 578	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10620	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 166	. 2 $/\$ $/\$ $/\$
79		2pos 8948	. . . . 5
80	29, 79	pm3.2i 270	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8777	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1228	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 146	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1343

wcel 2136 class class class wbr 3982

cfv 5188 (class class class)co 5842

cc 7751

cr 7752

cc0 7753

caddc 7756

cmul 7758

clt 7933

cle 7934

cmin 8069

cdiv 8568

c2 8908

c4 8910

cexp 10454

csqrt 10938

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-rp 9590 df-seqfrec 10381 df-exp 10455 df-rsqrt 10940

This theorem is referenced by: (None)

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