amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11265

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 9055	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 8002	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8640	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11176	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 8050	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10675	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10709	. . . . . . 7
13	12	oveq1i 5929	. . . . . 6
14		resqrtth 11178	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5935	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2238	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2226	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8402	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10774	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 8050	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 8050	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10725	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10727	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5937	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10773	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 9054	. . . . . . . . . . . 12
30		remulcl 8002	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 8051	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 8050	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8402	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 8050	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10773	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 8050	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8370	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 8050	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9228	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9139	. . . . . . . . . . 11
42	41	oveq1i 5929	. . . . . . . . . 10
43		2cnd 9057	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 8050	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 8045	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2240	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 8050	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8371	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2237	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2230	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 8051	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10773	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 8050	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10773	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 8050	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 9061	. . . . . . . . . 10
57		remulcl 8002	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 8050	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8226	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4058	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8556	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4052	. . 3 $/\$ $/\$ $/\$
67		remulcl 8002	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11180	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 9074	. . . . . 6
72		mulge0 8640	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8472	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 588	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10779	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 9075	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8903	. . 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 2164 class class class wbr 4030

cfv 5255 (class class class)co 5919

cc 7872

cr 7873

cc0 7874

caddc 7877

cmul 7879

clt 8056

cle 8057

cmin 8192

cdiv 8693

c2 9035

c4 9037

cexp 10612

csqrt 11143

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-rp 9723 df-seqfrec 10522 df-exp 10613 df-rsqrt 11145

This theorem is referenced by: (None)

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