amgm2 - Intuitionistic Logic Explorer

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Theorem amgm2 11111

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 8979	. . . . . 6
2		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
4		remulcl 7930	. . . . . . . . 9 $/\$
5	2, 3, 4	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
6		mulge0 8566	. . . . . . . 8 $/\$ $/\$ $/\$
7		resqrtcl 11022	. . . . . . . 8 $/\$
8	5, 6, 7	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
9	8	recnd 7976	. . . . . 6 $/\$ $/\$ $/\$
10		sqmul 10568	. . . . . 6 $/\$
11	1, 9, 10	sylancr 414	. . . . 5 $/\$ $/\$ $/\$
12		sq2 10601	. . . . . . 7
13	12	oveq1i 5879	. . . . . 6
14		resqrtth 11024	. . . . . . . 8 $/\$
15	5, 6, 14	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
16	15	oveq2d 5885	. . . . . 6 $/\$ $/\$ $/\$
17	13, 16	eqtrid 2222	. . . . 5 $/\$ $/\$ $/\$
18	11, 17	eqtrd 2210	. . . 4 $/\$ $/\$ $/\$
19	2, 3	resubcld 8328	. . . . . . 7 $/\$ $/\$ $/\$
20	19	sqge0d 10666	. . . . . 6 $/\$ $/\$ $/\$
21	2	recnd 7976	. . . . . . . . . 10 $/\$ $/\$ $/\$
22	3	recnd 7976	. . . . . . . . . 10 $/\$ $/\$ $/\$
23		binom2 10617	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
25		binom2sub 10619	. . . . . . . . . 10 $/\$
26	21, 22, 25	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
27	24, 26	oveq12d 5887	. . . . . . . 8 $/\$ $/\$ $/\$
28	2	resqcld 10665	. . . . . . . . . . 11 $/\$ $/\$ $/\$
29		2re 8978	. . . . . . . . . . . 12
30		remulcl 7930	. . . . . . . . . . . 12 $/\$
31	29, 5, 30	sylancr 414	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	28, 31	readdcld 7977	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	recnd 7976	. . . . . . . . 9 $/\$ $/\$ $/\$
34	28, 31	resubcld 8328	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	34	recnd 7976	. . . . . . . . 9 $/\$ $/\$ $/\$
36	3	resqcld 10665	. . . . . . . . . 10 $/\$ $/\$ $/\$
37	36	recnd 7976	. . . . . . . . 9 $/\$ $/\$ $/\$
38	33, 35, 37	pnpcan2d 8296	. . . . . . . 8 $/\$ $/\$ $/\$
39	31	recnd 7976	. . . . . . . . . 10 $/\$ $/\$ $/\$
40	39	2timesd 9150	. . . . . . . . 9 $/\$ $/\$ $/\$
41		2t2e4 9062	. . . . . . . . . . 11
42	41	oveq1i 5879	. . . . . . . . . 10
43		2cnd 8981	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	5	recnd 7976	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	43, 43, 44	mulassd 7971	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	42, 45	eqtr3id 2224	. . . . . . . . 9 $/\$ $/\$ $/\$
47	28	recnd 7976	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47, 39, 39	pnncand 8297	. . . . . . . . 9 $/\$ $/\$ $/\$
49	40, 46, 48	3eqtr4rd 2221	. . . . . . . 8 $/\$ $/\$ $/\$
50	27, 38, 49	3eqtrd 2214	. . . . . . 7 $/\$ $/\$ $/\$
51	2, 3	readdcld 7977	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	resqcld 10665	. . . . . . . . 9 $/\$ $/\$ $/\$
53	52	recnd 7976	. . . . . . . 8 $/\$ $/\$ $/\$
54	19	resqcld 10665	. . . . . . . . 9 $/\$ $/\$ $/\$
55	54	recnd 7976	. . . . . . . 8 $/\$ $/\$ $/\$
56		4re 8985	. . . . . . . . . 10
57		remulcl 7930	. . . . . . . . . 10 $/\$
58	56, 5, 57	sylancr 414	. . . . . . . . 9 $/\$ $/\$ $/\$
59	58	recnd 7976	. . . . . . . 8 $/\$ $/\$ $/\$
60		subsub23 8152	. . . . . . . 8 $/\$ $/\$
61	53, 55, 59, 60	syl3anc 1238	. . . . . . 7 $/\$ $/\$ $/\$
62	50, 61	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
63	20, 62	breqtrrd 4028	. . . . 5 $/\$ $/\$ $/\$
64	52, 58	subge0d 8482	. . . . 5 $/\$ $/\$ $/\$
65	63, 64	mpbid 147	. . . 4 $/\$ $/\$ $/\$
66	18, 65	eqbrtrd 4022	. . 3 $/\$ $/\$ $/\$
67		remulcl 7930	. . . . 5 $/\$
68	29, 8, 67	sylancr 414	. . . 4 $/\$ $/\$ $/\$
69		sqrtge0 11026	. . . . . 6 $/\$
70	5, 6, 69	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
71		0le2 8998	. . . . . 6
72		mulge0 8566	. . . . . 6 $/\$ $/\$ $/\$
73	29, 71, 72	mpanl12 436	. . . . 5 $/\$
74	8, 70, 73	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
75		addge0 8398	. . . . 5 $/\$ $/\$ $/\$
76	75	an4s 588	. . . 4 $/\$ $/\$ $/\$
77	68, 51, 74, 76	le2sqd 10671	. . 3 $/\$ $/\$ $/\$
78	66, 77	mpbird 167	. 2 $/\$ $/\$ $/\$
79		2pos 8999	. . . . 5
80	29, 79	pm3.2i 272	. . . 4 $/\$
81	80	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$
82		lemuldiv2 8828	. . 3 $/\$ $/\$ $/\$
83	8, 51, 81, 82	syl3anc 1238	. 2 $/\$ $/\$ $/\$
84	78, 83	mpbid 147	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148 class class class wbr 4000

cfv 5212 (class class class)co 5869

cc 7800

cr 7801

cc0 7802

caddc 7805

cmul 7807

clt 7982

cle 7983

cmin 8118

cdiv 8618

c2 8959

c4 8961

cexp 10505

csqrt 10989

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-rp 9641 df-seqfrec 10432 df-exp 10506 df-rsqrt 10991

This theorem is referenced by: (None)

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