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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > amgm4d | Structured version Visualization version GIF version |
Description: Arithmetic-geometric mean inequality for 𝑛 = 4. (Contributed by Stanislas Polu, 11-Sep-2020.) |
Ref | Expression |
---|---|
amgm4d.0 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
amgm4d.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
amgm4d.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
amgm4d.3 | ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
Ref | Expression |
---|---|
amgm4d | ⊢ (𝜑 → ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4)) ≤ ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 14011 | . . . 4 ⊢ (0..^4) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^4) ∈ Fin) |
4 | 4nn 12346 | . . . . 5 ⊢ 4 ∈ ℕ | |
5 | lbfzo0 13735 | . . . . 5 ⊢ (0 ∈ (0..^4) ↔ 4 ∈ ℕ) | |
6 | 4, 5 | mpbir 231 | . . . 4 ⊢ 0 ∈ (0..^4) |
7 | ne0i 4346 | . . . 4 ⊢ (0 ∈ (0..^4) → (0..^4) ≠ ∅) | |
8 | 6, 7 | mp1i 13 | . . 3 ⊢ (𝜑 → (0..^4) ≠ ∅) |
9 | amgm4d.0 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | amgm4d.1 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
11 | amgm4d.2 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
12 | amgm4d.3 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ+) | |
13 | 9, 10, 11, 12 | s4cld 14908 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+) |
14 | wrdf 14553 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) |
16 | s4len 14934 | . . . . . . 7 ⊢ (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4) |
18 | 17 | oveq2d 7446 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉)) = (0..^4)) |
19 | 18 | feq2d 6722 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+ ↔ 〈“𝐴𝐵𝐶𝐷”〉:(0..^4)⟶ℝ+)) |
20 | 15, 19 | mpbid 232 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉:(0..^4)⟶ℝ+) |
21 | 1, 3, 8, 20 | amgmlem 27047 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) ≤ ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4)))) |
22 | cnring 21420 | . . . . 5 ⊢ ℂfld ∈ Ring | |
23 | 1 | ringmgp 20256 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
24 | 22, 23 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
25 | 9 | rpcnd 13076 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
26 | 10 | rpcnd 13076 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
27 | 11 | rpcnd 13076 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
28 | 12 | rpcnd 13076 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
29 | 27, 28 | jca 511 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) |
30 | 25, 26, 29 | jca32 515 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)))) |
31 | cnfldbas 21385 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
32 | 1, 31 | mgpbas 20157 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
33 | cnfldmul 21389 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
34 | 1, 33 | mgpplusg 20155 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
35 | 32, 34 | gsumws4 44186 | . . . 4 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)))) → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 · (𝐵 · (𝐶 · 𝐷)))) |
36 | 24, 30, 35 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 · (𝐵 · (𝐶 · 𝐷)))) |
37 | 4nn0 12542 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
38 | hashfzo0 14465 | . . . . 5 ⊢ (4 ∈ ℕ0 → (♯‘(0..^4)) = 4) | |
39 | 37, 38 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^4)) = 4) |
40 | 39 | oveq2d 7446 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^4))) = (1 / 4)) |
41 | 36, 40 | oveq12d 7448 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) = ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4))) |
42 | ringmnd 20260 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
43 | 22, 42 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
44 | cnfldadd 21387 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
45 | 31, 44 | gsumws4 44186 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)))) → (ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 + (𝐵 + (𝐶 + 𝐷)))) |
46 | 43, 30, 45 | syl2anc 584 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 + (𝐵 + (𝐶 + 𝐷)))) |
47 | 46, 39 | oveq12d 7448 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4))) = ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) |
48 | 21, 41, 47 | 3brtr3d 5178 | 1 ⊢ (𝜑 → ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4)) ≤ ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∅c0 4338 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ≤ cle 11293 / cdiv 11917 ℕcn 12263 4c4 12320 ℕ0cn0 12523 ℝ+crp 13031 ..^cfzo 13690 ♯chash 14365 Word cword 14548 〈“cs4 14878 Σg cgsu 17486 Mndcmnd 18759 mulGrpcmgp 20151 Ringcrg 20250 ℂfldccnfld 21381 ↑𝑐ccxp 26611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-s4 14885 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-mulg 19098 df-subg 19153 df-ghm 19243 df-gim 19289 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-subrng 20562 df-subrg 20586 df-drng 20747 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-refld 21640 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 df-cxp 26613 |
This theorem is referenced by: (None) |
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