amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11283

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 9061	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8007	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8646	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11194	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8055	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10693	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10727	. . . . . . 7
13	12	oveq1i 5932	. . . . . 6
14		resqrtth 11196	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5938	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2241	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2229	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8407	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10792	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8055	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8055	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10743	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10745	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5940	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10791	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9060	. . . . . . . . . . . 12
30		remulcl 8007	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8056	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8055	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8407	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8055	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10791	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8055	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8375	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8055	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9234	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9145	. . . . . . . . . . 11
42	41	oveq1i 5932	. . . . . . . . . 10
43		2cnd 9063	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8055	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8050	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2243	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8055	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8376	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2240	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2233	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8056	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10791	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8055	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10791	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8055	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9067	. . . . . . . . . 10
57		remulcl 8007	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8055	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8231	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4061	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8562	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4055	. . 3 $/\$ $/\$ $/\$
67		remulcl 8007	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11198	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9080	. . . . . 6
72		mulge0 8646	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8478	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 588	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10797	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9081	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8909	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1249	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2167 class class class wbr 4033

cfv 5258 (class class class)co 5922

cc 7877

cr 7878

cc0 7879

caddc 7882

cmul 7884

clt 8061

cle 8062

cmin 8197

cdiv 8699

c2 9041

c4 9043

cexp 10630

csqrt 11161

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-rp 9729 df-seqfrec 10540 df-exp 10631 df-rsqrt 11163

This theorem is referenced by: (None)

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