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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 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 2798 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13337 | . . . 4 ⊢ (0..^3) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^3) ∈ Fin) |
4 | 3nn 11704 | . . . . 5 ⊢ 3 ∈ ℕ | |
5 | lbfzo0 13072 | . . . . 5 ⊢ (0 ∈ (0..^3) ↔ 3 ∈ ℕ) | |
6 | 4, 5 | mpbir 234 | . . . 4 ⊢ 0 ∈ (0..^3) |
7 | ne0i 4250 | . . . 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 14225 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ Word ℝ+) |
13 | wrdf 13862 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶”〉:(0..^(♯‘〈“𝐴𝐵𝐶”〉))⟶ℝ+) | |
14 | s3len 14247 | . . . . . . . . 9 ⊢ (♯‘〈“𝐴𝐵𝐶”〉) = 3 | |
15 | df-3 11689 | . . . . . . . . 9 ⊢ 3 = (2 + 1) | |
16 | 14, 15 | eqtri 2821 | . . . . . . . 8 ⊢ (♯‘〈“𝐴𝐵𝐶”〉) = (2 + 1) |
17 | 16 | oveq2i 7146 | . . . . . . 7 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = (0..^(2 + 1)) |
18 | 17 | feq2i 6479 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉:(0..^(♯‘〈“𝐴𝐵𝐶”〉))⟶ℝ+ ↔ 〈“𝐴𝐵𝐶”〉:(0..^(2 + 1))⟶ℝ+) |
19 | 13, 18 | sylib 221 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶”〉:(0..^(2 + 1))⟶ℝ+) |
20 | 15 | oveq2i 7146 | . . . . . 6 ⊢ (0..^3) = (0..^(2 + 1)) |
21 | 20 | feq2i 6479 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉:(0..^3)⟶ℝ+ ↔ 〈“𝐴𝐵𝐶”〉:(0..^(2 + 1))⟶ℝ+) |
22 | 19, 21 | sylibr 237 | . . . 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 25575 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶”〉)↑𝑐(1 / (♯‘(0..^3)))) ≤ ((ℂfld Σg 〈“𝐴𝐵𝐶”〉) / (♯‘(0..^3)))) |
25 | cnring 20113 | . . . . 5 ⊢ ℂfld ∈ Ring | |
26 | 1 | ringmgp 19296 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
27 | 25, 26 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
28 | 9 | rpcnd 12421 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
29 | 10 | rpcnd 12421 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
30 | 11 | rpcnd 12421 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
31 | 28, 29, 30 | jca32 519 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))) |
32 | cnfldbas 20095 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
33 | 1, 32 | mgpbas 19238 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
34 | cnfldmul 20097 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
35 | 1, 34 | mgpplusg 19236 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
36 | 33, 35 | gsumws3 40902 | . . . 4 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))) → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶”〉) = (𝐴 · (𝐵 · 𝐶))) |
37 | 27, 31, 36 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶”〉) = (𝐴 · (𝐵 · 𝐶))) |
38 | 3nn0 11903 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
39 | hashfzo0 13787 | . . . . 5 ⊢ (3 ∈ ℕ0 → (♯‘(0..^3)) = 3) | |
40 | 38, 39 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^3)) = 3) |
41 | 40 | oveq2d 7151 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^3))) = (1 / 3)) |
42 | 37, 41 | oveq12d 7153 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶”〉)↑𝑐(1 / (♯‘(0..^3)))) = ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3))) |
43 | ringmnd 19300 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
44 | 25, 43 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
45 | cnfldadd 20096 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
46 | 32, 45 | gsumws3 40902 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))) → (ℂfld Σg 〈“𝐴𝐵𝐶”〉) = (𝐴 + (𝐵 + 𝐶))) |
47 | 44, 31, 46 | syl2anc 587 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝐴𝐵𝐶”〉) = (𝐴 + (𝐵 + 𝐶))) |
48 | 47, 40 | oveq12d 7153 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵𝐶”〉) / (♯‘(0..^3))) = ((𝐴 + (𝐵 + 𝐶)) / 3)) |
49 | 24, 42, 48 | 3brtr3d 5061 | 1 ⊢ (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 / cdiv 11286 ℕcn 11625 2c2 11680 3c3 11681 ℕ0cn0 11885 ℝ+crp 12377 ..^cfzo 13028 ♯chash 13686 Word cword 13857 〈“cs3 14195 Σg cgsu 16706 Mndcmnd 17903 mulGrpcmgp 19232 Ringcrg 19290 ℂfldccnfld 20091 ↑𝑐ccxp 25147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-ghm 18348 df-gim 18391 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-subrg 19526 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-refld 20294 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-cxp 25149 |
This theorem is referenced by: (None) |
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