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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 2726 | . . 3 β’ (mulGrpββfld) = (mulGrpββfld) | |
2 | fzofi 13942 | . . . 4 β’ (0..^3) β Fin | |
3 | 2 | a1i 11 | . . 3 β’ (π β (0..^3) β Fin) |
4 | 3nn 12292 | . . . . 5 β’ 3 β β | |
5 | lbfzo0 13675 | . . . . 5 β’ (0 β (0..^3) β 3 β β) | |
6 | 4, 5 | mpbir 230 | . . . 4 β’ 0 β (0..^3) |
7 | ne0i 4329 | . . . 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 14827 | . . . 4 β’ (π β β¨βπ΄π΅πΆββ© β Word β+) |
13 | wrdf 14473 | . . . . . 6 β’ (β¨βπ΄π΅πΆββ© β Word β+ β β¨βπ΄π΅πΆββ©:(0..^(β―ββ¨βπ΄π΅πΆββ©))βΆβ+) | |
14 | s3len 14849 | . . . . . . . . 9 β’ (β―ββ¨βπ΄π΅πΆββ©) = 3 | |
15 | df-3 12277 | . . . . . . . . 9 β’ 3 = (2 + 1) | |
16 | 14, 15 | eqtri 2754 | . . . . . . . 8 β’ (β―ββ¨βπ΄π΅πΆββ©) = (2 + 1) |
17 | 16 | oveq2i 7415 | . . . . . . 7 β’ (0..^(β―ββ¨βπ΄π΅πΆββ©)) = (0..^(2 + 1)) |
18 | 17 | feq2i 6702 | . . . . . 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 7415 | . . . . . 6 β’ (0..^3) = (0..^(2 + 1)) |
21 | 20 | feq2i 6702 | . . . . 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 26873 | . 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 21275 | . . . . 5 β’ βfld β Ring | |
26 | 1 | ringmgp 20142 | . . . . 5 β’ (βfld β Ring β (mulGrpββfld) β Mnd) |
27 | 25, 26 | mp1i 13 | . . . 4 β’ (π β (mulGrpββfld) β Mnd) |
28 | 9 | rpcnd 13021 | . . . . 5 β’ (π β π΄ β β) |
29 | 10 | rpcnd 13021 | . . . . 5 β’ (π β π΅ β β) |
30 | 11 | rpcnd 13021 | . . . . 5 β’ (π β πΆ β β) |
31 | 28, 29, 30 | jca32 515 | . . . 4 β’ (π β (π΄ β β β§ (π΅ β β β§ πΆ β β))) |
32 | cnfldbas 21240 | . . . . . 6 β’ β = (Baseββfld) | |
33 | 1, 32 | mgpbas 20043 | . . . . 5 β’ β = (Baseβ(mulGrpββfld)) |
34 | cnfldmul 21244 | . . . . . 6 β’ Β· = (.rββfld) | |
35 | 1, 34 | mgpplusg 20041 | . . . . 5 β’ Β· = (+gβ(mulGrpββfld)) |
36 | 33, 35 | gsumws3 43505 | . . . 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 583 | . . 3 β’ (π β ((mulGrpββfld) Ξ£g β¨βπ΄π΅πΆββ©) = (π΄ Β· (π΅ Β· πΆ))) |
38 | 3nn0 12491 | . . . . 5 β’ 3 β β0 | |
39 | hashfzo0 14393 | . . . . 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 7420 | . . 3 β’ (π β (1 / (β―β(0..^3))) = (1 / 3)) |
42 | 37, 41 | oveq12d 7422 | . 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 20146 | . . . . 5 β’ (βfld β Ring β βfld β Mnd) | |
44 | 25, 43 | mp1i 13 | . . . 4 β’ (π β βfld β Mnd) |
45 | cnfldadd 21242 | . . . . 5 β’ + = (+gββfld) | |
46 | 32, 45 | gsumws3 43505 | . . . 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 583 | . . 3 β’ (π β (βfld Ξ£g β¨βπ΄π΅πΆββ©) = (π΄ + (π΅ + πΆ))) |
48 | 47, 40 | oveq12d 7422 | . 2 β’ (π β ((βfld Ξ£g β¨βπ΄π΅πΆββ©) / (β―β(0..^3))) = ((π΄ + (π΅ + πΆ)) / 3)) |
49 | 24, 42, 48 | 3brtr3d 5172 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 β c0 4317 class class class wbr 5141 βΆwf 6532 βcfv 6536 (class class class)co 7404 Fincfn 8938 βcc 11107 0cc0 11109 1c1 11110 + caddc 11112 Β· cmul 11114 β€ cle 11250 / cdiv 11872 βcn 12213 2c2 12268 3c3 12269 β0cn0 12473 β+crp 12977 ..^cfzo 13630 β―chash 14293 Word cword 14468 β¨βcs3 14797 Ξ£g cgsu 17393 Mndcmnd 18665 mulGrpcmgp 20037 Ringcrg 20136 βfldccnfld 21236 βπccxp 26440 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-mulg 18994 df-subg 19048 df-ghm 19137 df-gim 19182 df-cntz 19231 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-dvr 20301 df-subrng 20444 df-subrg 20469 df-drng 20587 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-refld 21494 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-cmp 23242 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-limc 25746 df-dv 25747 df-log 26441 df-cxp 26442 |
This theorem is referenced by: (None) |
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