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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > amgm3d | Structured version Visualization version GIF version |
Description: Arithmetic-geometric mean inequality for π = 3. (Contributed by Stanislas Polu, 11-Sep-2020.) |
Ref | Expression |
---|---|
amgm3d.0 | β’ (π β π΄ β β+) |
amgm3d.1 | β’ (π β π΅ β β+) |
amgm3d.2 | β’ (π β πΆ β β+) |
Ref | Expression |
---|---|
amgm3d | β’ (π β ((π΄ Β· (π΅ Β· πΆ))βπ(1 / 3)) β€ ((π΄ + (π΅ + πΆ)) / 3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . 3 β’ (mulGrpββfld) = (mulGrpββfld) | |
2 | fzofi 13977 | . . . 4 β’ (0..^3) β Fin | |
3 | 2 | a1i 11 | . . 3 β’ (π β (0..^3) β Fin) |
4 | 3nn 12327 | . . . . 5 β’ 3 β β | |
5 | lbfzo0 13710 | . . . . 5 β’ (0 β (0..^3) β 3 β β) | |
6 | 4, 5 | mpbir 230 | . . . 4 β’ 0 β (0..^3) |
7 | ne0i 4336 | . . . 4 β’ (0 β (0..^3) β (0..^3) β β ) | |
8 | 6, 7 | mp1i 13 | . . 3 β’ (π β (0..^3) β β ) |
9 | amgm3d.0 | . . . . 5 β’ (π β π΄ β β+) | |
10 | amgm3d.1 | . . . . 5 β’ (π β π΅ β β+) | |
11 | amgm3d.2 | . . . . 5 β’ (π β πΆ β β+) | |
12 | 9, 10, 11 | s3cld 14861 | . . . 4 β’ (π β β¨βπ΄π΅πΆββ© β Word β+) |
13 | wrdf 14507 | . . . . . 6 β’ (β¨βπ΄π΅πΆββ© β Word β+ β β¨βπ΄π΅πΆββ©:(0..^(β―ββ¨βπ΄π΅πΆββ©))βΆβ+) | |
14 | s3len 14883 | . . . . . . . . 9 β’ (β―ββ¨βπ΄π΅πΆββ©) = 3 | |
15 | df-3 12312 | . . . . . . . . 9 β’ 3 = (2 + 1) | |
16 | 14, 15 | eqtri 2755 | . . . . . . . 8 β’ (β―ββ¨βπ΄π΅πΆββ©) = (2 + 1) |
17 | 16 | oveq2i 7435 | . . . . . . 7 β’ (0..^(β―ββ¨βπ΄π΅πΆββ©)) = (0..^(2 + 1)) |
18 | 17 | feq2i 6717 | . . . . . 6 β’ (β¨βπ΄π΅πΆββ©:(0..^(β―ββ¨βπ΄π΅πΆββ©))βΆβ+ β β¨βπ΄π΅πΆββ©:(0..^(2 + 1))βΆβ+) |
19 | 13, 18 | sylib 217 | . . . . 5 β’ (β¨βπ΄π΅πΆββ© β Word β+ β β¨βπ΄π΅πΆββ©:(0..^(2 + 1))βΆβ+) |
20 | 15 | oveq2i 7435 | . . . . . 6 β’ (0..^3) = (0..^(2 + 1)) |
21 | 20 | feq2i 6717 | . . . . 5 β’ (β¨βπ΄π΅πΆββ©:(0..^3)βΆβ+ β β¨βπ΄π΅πΆββ©:(0..^(2 + 1))βΆβ+) |
22 | 19, 21 | sylibr 233 | . . . 4 β’ (β¨βπ΄π΅πΆββ© β Word β+ β β¨βπ΄π΅πΆββ©:(0..^3)βΆβ+) |
23 | 12, 22 | syl 17 | . . 3 β’ (π β β¨βπ΄π΅πΆββ©:(0..^3)βΆβ+) |
24 | 1, 3, 8, 23 | amgmlem 26940 | . 2 β’ (π β (((mulGrpββfld) Ξ£g β¨βπ΄π΅πΆββ©)βπ(1 / (β―β(0..^3)))) β€ ((βfld Ξ£g β¨βπ΄π΅πΆββ©) / (β―β(0..^3)))) |
25 | cnring 21323 | . . . . 5 β’ βfld β Ring | |
26 | 1 | ringmgp 20184 | . . . . 5 β’ (βfld β Ring β (mulGrpββfld) β Mnd) |
27 | 25, 26 | mp1i 13 | . . . 4 β’ (π β (mulGrpββfld) β Mnd) |
28 | 9 | rpcnd 13056 | . . . . 5 β’ (π β π΄ β β) |
29 | 10 | rpcnd 13056 | . . . . 5 β’ (π β π΅ β β) |
30 | 11 | rpcnd 13056 | . . . . 5 β’ (π β πΆ β β) |
31 | 28, 29, 30 | jca32 514 | . . . 4 β’ (π β (π΄ β β β§ (π΅ β β β§ πΆ β β))) |
32 | cnfldbas 21288 | . . . . . 6 β’ β = (Baseββfld) | |
33 | 1, 32 | mgpbas 20085 | . . . . 5 β’ β = (Baseβ(mulGrpββfld)) |
34 | cnfldmul 21292 | . . . . . 6 β’ Β· = (.rββfld) | |
35 | 1, 34 | mgpplusg 20083 | . . . . 5 β’ Β· = (+gβ(mulGrpββfld)) |
36 | 33, 35 | gsumws3 43629 | . . . 4 β’ (((mulGrpββfld) β Mnd β§ (π΄ β β β§ (π΅ β β β§ πΆ β β))) β ((mulGrpββfld) Ξ£g β¨βπ΄π΅πΆββ©) = (π΄ Β· (π΅ Β· πΆ))) |
37 | 27, 31, 36 | syl2anc 582 | . . 3 β’ (π β ((mulGrpββfld) Ξ£g β¨βπ΄π΅πΆββ©) = (π΄ Β· (π΅ Β· πΆ))) |
38 | 3nn0 12526 | . . . . 5 β’ 3 β β0 | |
39 | hashfzo0 14427 | . . . . 5 β’ (3 β β0 β (β―β(0..^3)) = 3) | |
40 | 38, 39 | mp1i 13 | . . . 4 β’ (π β (β―β(0..^3)) = 3) |
41 | 40 | oveq2d 7440 | . . 3 β’ (π β (1 / (β―β(0..^3))) = (1 / 3)) |
42 | 37, 41 | oveq12d 7442 | . 2 β’ (π β (((mulGrpββfld) Ξ£g β¨βπ΄π΅πΆββ©)βπ(1 / (β―β(0..^3)))) = ((π΄ Β· (π΅ Β· πΆ))βπ(1 / 3))) |
43 | ringmnd 20188 | . . . . 5 β’ (βfld β Ring β βfld β Mnd) | |
44 | 25, 43 | mp1i 13 | . . . 4 β’ (π β βfld β Mnd) |
45 | cnfldadd 21290 | . . . . 5 β’ + = (+gββfld) | |
46 | 32, 45 | gsumws3 43629 | . . . 4 β’ ((βfld β Mnd β§ (π΄ β β β§ (π΅ β β β§ πΆ β β))) β (βfld Ξ£g β¨βπ΄π΅πΆββ©) = (π΄ + (π΅ + πΆ))) |
47 | 44, 31, 46 | syl2anc 582 | . . 3 β’ (π β (βfld Ξ£g β¨βπ΄π΅πΆββ©) = (π΄ + (π΅ + πΆ))) |
48 | 47, 40 | oveq12d 7442 | . 2 β’ (π β ((βfld Ξ£g β¨βπ΄π΅πΆββ©) / (β―β(0..^3))) = ((π΄ + (π΅ + πΆ)) / 3)) |
49 | 24, 42, 48 | 3brtr3d 5181 | 1 β’ (π β ((π΄ Β· (π΅ Β· πΆ))βπ(1 / 3)) β€ ((π΄ + (π΅ + πΆ)) / 3)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2936 β c0 4324 class class class wbr 5150 βΆwf 6547 βcfv 6551 (class class class)co 7424 Fincfn 8968 βcc 11142 0cc0 11144 1c1 11145 + caddc 11147 Β· cmul 11149 β€ cle 11285 / cdiv 11907 βcn 12248 2c2 12303 3c3 12304 β0cn0 12508 β+crp 13012 ..^cfzo 13665 β―chash 14327 Word cword 14502 β¨βcs3 14831 Ξ£g cgsu 17427 Mndcmnd 18699 mulGrpcmgp 20079 Ringcrg 20178 βfldccnfld 21284 βπccxp 26507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223 ax-mulf 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-fl 13795 df-mod 13873 df-seq 14005 df-exp 14065 df-fac 14271 df-bc 14300 df-hash 14328 df-word 14503 df-concat 14559 df-s1 14584 df-s2 14837 df-s3 14838 df-shft 15052 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-limsup 15453 df-clim 15470 df-rlim 15471 df-sum 15671 df-ef 16049 df-sin 16051 df-cos 16052 df-pi 16054 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-submnd 18746 df-grp 18898 df-minusg 18899 df-mulg 19029 df-subg 19083 df-ghm 19173 df-gim 19218 df-cntz 19273 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-cring 20181 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-dvr 20345 df-subrng 20488 df-subrg 20513 df-drng 20631 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-refld 21542 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-cmp 23309 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-limc 25813 df-dv 25814 df-log 26508 df-cxp 26509 |
This theorem is referenced by: (None) |
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