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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > amgm2d | Structured version Visualization version GIF version |
Description: Arithmetic-geometric mean inequality for π = 2, derived from amgmlem 26355. (Contributed by Stanislas Polu, 8-Sep-2020.) |
Ref | Expression |
---|---|
amgm2d.0 | β’ (π β π΄ β β+) |
amgm2d.1 | β’ (π β π΅ β β+) |
Ref | Expression |
---|---|
amgm2d | β’ (π β ((π΄ Β· π΅)βπ(1 / 2)) β€ ((π΄ + π΅) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (mulGrpββfld) = (mulGrpββfld) | |
2 | fzofi 13885 | . . . 4 β’ (0..^2) β Fin | |
3 | 2 | a1i 11 | . . 3 β’ (π β (0..^2) β Fin) |
4 | 2nn 12231 | . . . . . 6 β’ 2 β β | |
5 | lbfzo0 13618 | . . . . . 6 β’ (0 β (0..^2) β 2 β β) | |
6 | 4, 5 | mpbir 230 | . . . . 5 β’ 0 β (0..^2) |
7 | 6 | ne0ii 4298 | . . . 4 β’ (0..^2) β β |
8 | 7 | a1i 11 | . . 3 β’ (π β (0..^2) β β ) |
9 | amgm2d.0 | . . . . 5 β’ (π β π΄ β β+) | |
10 | amgm2d.1 | . . . . 5 β’ (π β π΅ β β+) | |
11 | 9, 10 | s2cld 14766 | . . . 4 β’ (π β β¨βπ΄π΅ββ© β Word β+) |
12 | wrdf 14413 | . . . . 5 β’ (β¨βπ΄π΅ββ© β Word β+ β β¨βπ΄π΅ββ©:(0..^(β―ββ¨βπ΄π΅ββ©))βΆβ+) | |
13 | s2len 14784 | . . . . . . . 8 β’ (β―ββ¨βπ΄π΅ββ©) = 2 | |
14 | 13 | eqcomi 2742 | . . . . . . 7 β’ 2 = (β―ββ¨βπ΄π΅ββ©) |
15 | 14 | oveq2i 7369 | . . . . . 6 β’ (0..^2) = (0..^(β―ββ¨βπ΄π΅ββ©)) |
16 | 15 | feq2i 6661 | . . . . 5 β’ (β¨βπ΄π΅ββ©:(0..^2)βΆβ+ β β¨βπ΄π΅ββ©:(0..^(β―ββ¨βπ΄π΅ββ©))βΆβ+) |
17 | 12, 16 | sylibr 233 | . . . 4 β’ (β¨βπ΄π΅ββ© β Word β+ β β¨βπ΄π΅ββ©:(0..^2)βΆβ+) |
18 | 11, 17 | syl 17 | . . 3 β’ (π β β¨βπ΄π΅ββ©:(0..^2)βΆβ+) |
19 | 1, 3, 8, 18 | amgmlem 26355 | . 2 β’ (π β (((mulGrpββfld) Ξ£g β¨βπ΄π΅ββ©)βπ(1 / (β―β(0..^2)))) β€ ((βfld Ξ£g β¨βπ΄π΅ββ©) / (β―β(0..^2)))) |
20 | cnring 20835 | . . . . 5 β’ βfld β Ring | |
21 | 1 | ringmgp 19975 | . . . . 5 β’ (βfld β Ring β (mulGrpββfld) β Mnd) |
22 | 20, 21 | mp1i 13 | . . . 4 β’ (π β (mulGrpββfld) β Mnd) |
23 | 9 | rpcnd 12964 | . . . 4 β’ (π β π΄ β β) |
24 | 10 | rpcnd 12964 | . . . 4 β’ (π β π΅ β β) |
25 | cnfldbas 20816 | . . . . . 6 β’ β = (Baseββfld) | |
26 | 1, 25 | mgpbas 19907 | . . . . 5 β’ β = (Baseβ(mulGrpββfld)) |
27 | cnfldmul 20818 | . . . . . 6 β’ Β· = (.rββfld) | |
28 | 1, 27 | mgpplusg 19905 | . . . . 5 β’ Β· = (+gβ(mulGrpββfld)) |
29 | 26, 28 | gsumws2 18657 | . . . 4 β’ (((mulGrpββfld) β Mnd β§ π΄ β β β§ π΅ β β) β ((mulGrpββfld) Ξ£g β¨βπ΄π΅ββ©) = (π΄ Β· π΅)) |
30 | 22, 23, 24, 29 | syl3anc 1372 | . . 3 β’ (π β ((mulGrpββfld) Ξ£g β¨βπ΄π΅ββ©) = (π΄ Β· π΅)) |
31 | 2nn0 12435 | . . . . 5 β’ 2 β β0 | |
32 | hashfzo0 14336 | . . . . 5 β’ (2 β β0 β (β―β(0..^2)) = 2) | |
33 | 31, 32 | mp1i 13 | . . . 4 β’ (π β (β―β(0..^2)) = 2) |
34 | 33 | oveq2d 7374 | . . 3 β’ (π β (1 / (β―β(0..^2))) = (1 / 2)) |
35 | 30, 34 | oveq12d 7376 | . 2 β’ (π β (((mulGrpββfld) Ξ£g β¨βπ΄π΅ββ©)βπ(1 / (β―β(0..^2)))) = ((π΄ Β· π΅)βπ(1 / 2))) |
36 | ringmnd 19979 | . . . . 5 β’ (βfld β Ring β βfld β Mnd) | |
37 | 20, 36 | mp1i 13 | . . . 4 β’ (π β βfld β Mnd) |
38 | cnfldadd 20817 | . . . . 5 β’ + = (+gββfld) | |
39 | 25, 38 | gsumws2 18657 | . . . 4 β’ ((βfld β Mnd β§ π΄ β β β§ π΅ β β) β (βfld Ξ£g β¨βπ΄π΅ββ©) = (π΄ + π΅)) |
40 | 37, 23, 24, 39 | syl3anc 1372 | . . 3 β’ (π β (βfld Ξ£g β¨βπ΄π΅ββ©) = (π΄ + π΅)) |
41 | 40, 33 | oveq12d 7376 | . 2 β’ (π β ((βfld Ξ£g β¨βπ΄π΅ββ©) / (β―β(0..^2))) = ((π΄ + π΅) / 2)) |
42 | 19, 35, 41 | 3brtr3d 5137 | 1 β’ (π β ((π΄ Β· π΅)βπ(1 / 2)) β€ ((π΄ + π΅) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wne 2940 β c0 4283 class class class wbr 5106 βΆwf 6493 βcfv 6497 (class class class)co 7358 Fincfn 8886 βcc 11054 0cc0 11056 1c1 11057 + caddc 11059 Β· cmul 11061 β€ cle 11195 / cdiv 11817 βcn 12158 2c2 12213 β0cn0 12418 β+crp 12920 ..^cfzo 13573 β―chash 14236 Word cword 14408 β¨βcs2 14736 Ξ£g cgsu 17327 Mndcmnd 18561 mulGrpcmgp 19901 Ringcrg 19969 βfldccnfld 20812 βπccxp 25927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ioc 13275 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-mod 13781 df-seq 13913 df-exp 13974 df-fac 14180 df-bc 14209 df-hash 14237 df-word 14409 df-concat 14465 df-s1 14490 df-s2 14743 df-shft 14958 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-limsup 15359 df-clim 15376 df-rlim 15377 df-sum 15577 df-ef 15955 df-sin 15957 df-cos 15958 df-pi 15960 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-rest 17309 df-topn 17310 df-0g 17328 df-gsum 17329 df-topgen 17330 df-pt 17331 df-prds 17334 df-xrs 17389 df-qtop 17394 df-imas 17395 df-xps 17397 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-mulg 18878 df-subg 18930 df-ghm 19011 df-gim 19054 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-subrg 20234 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-refld 21025 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-cmp 22754 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 df-cxp 25929 |
This theorem is referenced by: (None) |
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