amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 10004

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 8110	. . . . . 6
2		simpll 495	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 497	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 7101	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 403	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 7719	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 9915	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 403	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 7147	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 9538	. . . . . 6 $/\$
11	1, 9, 10	sylancr 405	. . . . 5 $/\$ $/\$ $/\$
12		sq2 9571	. . . . . . 7
13	12	oveq1i 5542	. . . . . 6
14		resqrtth 9917	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 403	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5548	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	syl5eq 2125	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2113	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 7485	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 9632	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 7147	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 7147	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 9585	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 403	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 9587	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 403	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5550	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 9631	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 8109	. . . . . . . . . . . 12
30		remulcl 7101	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 405	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 7148	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 7147	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 7485	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 7147	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 9631	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 7147	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 7457	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 7147	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 8273	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 8186	. . . . . . . . . . 11
42	41	oveq1i 5542	. . . . . . . . . 10
43		2cnd 8112	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 7147	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 7142	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	syl5eqr 2127	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 7147	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 7458	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2124	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2117	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 7148	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 9631	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 7147	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 9631	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 7147	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 8116	. . . . . . . . . 10
57		remulcl 7101	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 405	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 7147	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 7313	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1169	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 145	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 3811	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 7635	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 145	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 3805	. . 3 $/\$ $/\$ $/\$
67		remulcl 7101	. . . . 5 $/\$
68	29, 8, 67	sylancr 405	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 9919	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 403	. . . . 5 $/\$ $/\$ $/\$
71		0le2 8129	. . . . . 6
72		mulge0 7719	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 426	. . . . 5 $/\$
74	8, 70, 73	syl2anc 403	. . . 4 $/\$ $/\$ $/\$
75		addge0 7555	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 552	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 9637	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 165	. 2 $/\$ $/\$ $/\$
79		2pos 8130	. . . . 5
80	29, 79	pm3.2i 266	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 7960	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1169	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 145	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wb 103

wceq 1284

wcel 1433 class class class wbr 3785

cfv 4922 (class class class)co 5532

cc 6979

cr 6980

cc0 6981

caddc 6984

cmul 6986

clt 7153

cle 7154

cmin 7279

cdiv 7760

c2 8089

c4 8091

cexp 9475

csqrt 9882

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-rp 8735 df-iseq 9432 df-iexp 9476 df-rsqrt 9884

This theorem is referenced by: (None)

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