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 𝑛 = 3. (Contributed by Stanislas Polu, 11-Sep-2020.) |
Ref | Expression |
---|---|
amgm3d.0 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
amgm3d.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
amgm3d.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
Ref | Expression |
---|---|
amgm3d | ⊢ (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 12773 | . . . 4 ⊢ (0..^3) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^3) ∈ Fin) |
4 | 3nn 11186 | . . . . 5 ⊢ 3 ∈ ℕ | |
5 | lbfzo0 12507 | . . . . 5 ⊢ (0 ∈ (0..^3) ↔ 3 ∈ ℕ) | |
6 | 4, 5 | mpbir 221 | . . . 4 ⊢ 0 ∈ (0..^3) |
7 | ne0i 3921 | . . . 4 ⊢ (0 ∈ (0..^3) → (0..^3) ≠ ∅) | |
8 | 6, 7 | mp1i 13 | . . 3 ⊢ (𝜑 → (0..^3) ≠ ∅) |
9 | amgm3d.0 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | amgm3d.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
11 | amgm3d.2 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
12 | 9, 10, 11 | s3cld 13617 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ Word ℝ+) |
13 | wrdf 13310 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶”〉:(0..^(#‘〈“𝐴𝐵𝐶”〉))⟶ℝ+) | |
14 | s3len 13639 | . . . . . . . . 9 ⊢ (#‘〈“𝐴𝐵𝐶”〉) = 3 | |
15 | df-3 11080 | . . . . . . . . 9 ⊢ 3 = (2 + 1) | |
16 | 14, 15 | eqtri 2644 | . . . . . . . 8 ⊢ (#‘〈“𝐴𝐵𝐶”〉) = (2 + 1) |
17 | 16 | oveq2i 6661 | . . . . . . 7 ⊢ (0..^(#‘〈“𝐴𝐵𝐶”〉)) = (0..^(2 + 1)) |
18 | 17 | feq2i 6037 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉:(0..^(#‘〈“𝐴𝐵𝐶”〉))⟶ℝ+ ↔ 〈“𝐴𝐵𝐶”〉:(0..^(2 + 1))⟶ℝ+) |
19 | 13, 18 | sylib 208 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶”〉:(0..^(2 + 1))⟶ℝ+) |
20 | 15 | oveq2i 6661 | . . . . . 6 ⊢ (0..^3) = (0..^(2 + 1)) |
21 | 20 | feq2i 6037 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉:(0..^3)⟶ℝ+ ↔ 〈“𝐴𝐵𝐶”〉:(0..^(2 + 1))⟶ℝ+) |
22 | 19, 21 | sylibr 224 | . . . 4 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶”〉:(0..^3)⟶ℝ+) |
23 | 12, 22 | syl 17 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉:(0..^3)⟶ℝ+) |
24 | 1, 3, 8, 23 | amgmlem 24716 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶”〉)↑𝑐(1 / (#‘(0..^3)))) ≤ ((ℂfld Σg 〈“𝐴𝐵𝐶”〉) / (#‘(0..^3)))) |
25 | cnring 19768 | . . . . 5 ⊢ ℂfld ∈ Ring | |
26 | 1 | ringmgp 18553 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
27 | 25, 26 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
28 | 9 | rpcnd 11874 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
29 | 10 | rpcnd 11874 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
30 | 11 | rpcnd 11874 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
31 | 28, 29, 30 | jca32 558 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))) |
32 | cnfldbas 19750 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
33 | 1, 32 | mgpbas 18495 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
34 | cnfldmul 19752 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
35 | 1, 34 | mgpplusg 18493 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
36 | 33, 35 | gsumws3 38499 | . . . 4 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))) → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶”〉) = (𝐴 · (𝐵 · 𝐶))) |
37 | 27, 31, 36 | syl2anc 693 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶”〉) = (𝐴 · (𝐵 · 𝐶))) |
38 | 3nn0 11310 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
39 | hashfzo0 13217 | . . . . 5 ⊢ (3 ∈ ℕ0 → (#‘(0..^3)) = 3) | |
40 | 38, 39 | mp1i 13 | . . . 4 ⊢ (𝜑 → (#‘(0..^3)) = 3) |
41 | 40 | oveq2d 6666 | . . 3 ⊢ (𝜑 → (1 / (#‘(0..^3))) = (1 / 3)) |
42 | 37, 41 | oveq12d 6668 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶”〉)↑𝑐(1 / (#‘(0..^3)))) = ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3))) |
43 | ringmnd 18556 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
44 | 25, 43 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
45 | cnfldadd 19751 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
46 | 32, 45 | gsumws3 38499 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))) → (ℂfld Σg 〈“𝐴𝐵𝐶”〉) = (𝐴 + (𝐵 + 𝐶))) |
47 | 44, 31, 46 | syl2anc 693 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝐴𝐵𝐶”〉) = (𝐴 + (𝐵 + 𝐶))) |
48 | 47, 40 | oveq12d 6668 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵𝐶”〉) / (#‘(0..^3))) = ((𝐴 + (𝐵 + 𝐶)) / 3)) |
49 | 24, 42, 48 | 3brtr3d 4684 | 1 ⊢ (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ≤ cle 10075 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 ℕ0cn0 11292 ℝ+crp 11832 ..^cfzo 12465 #chash 13117 Word cword 13291 〈“cs3 13587 Σg cgsu 16101 Mndcmnd 17294 mulGrpcmgp 18489 Ringcrg 18547 ℂfldccnfld 19746 ↑𝑐ccxp 24302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-refld 19951 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 |
This theorem is referenced by: (None) |
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