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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 2760 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 12987 | . . . 4 ⊢ (0..^4) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^4) ∈ Fin) |
4 | 4nn 11399 | . . . . 5 ⊢ 4 ∈ ℕ | |
5 | lbfzo0 12722 | . . . . 5 ⊢ (0 ∈ (0..^4) ↔ 4 ∈ ℕ) | |
6 | 4, 5 | mpbir 221 | . . . 4 ⊢ 0 ∈ (0..^4) |
7 | ne0i 4064 | . . . 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 13838 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+) |
14 | wrdf 13516 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) |
16 | s4len 13864 | . . . . . . 7 ⊢ (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4) |
18 | 17 | oveq2d 6830 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉)) = (0..^4)) |
19 | 18 | feq2d 6192 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+ ↔ 〈“𝐴𝐵𝐶𝐷”〉:(0..^4)⟶ℝ+)) |
20 | 15, 19 | mpbid 222 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉:(0..^4)⟶ℝ+) |
21 | 1, 3, 8, 20 | amgmlem 24936 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) ≤ ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4)))) |
22 | cnring 19990 | . . . . 5 ⊢ ℂfld ∈ Ring | |
23 | 1 | ringmgp 18773 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
24 | 22, 23 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
25 | 9 | rpcnd 12087 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
26 | 10 | rpcnd 12087 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
27 | 11 | rpcnd 12087 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
28 | 12 | rpcnd 12087 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
29 | 27, 28 | jca 555 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) |
30 | 25, 26, 29 | jca32 559 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)))) |
31 | cnfldbas 19972 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
32 | 1, 31 | mgpbas 18715 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
33 | cnfldmul 19974 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
34 | 1, 33 | mgpplusg 18713 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
35 | 32, 34 | gsumws4 39020 | . . . 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 696 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 · (𝐵 · (𝐶 · 𝐷)))) |
37 | 4nn0 11523 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
38 | hashfzo0 13429 | . . . . 5 ⊢ (4 ∈ ℕ0 → (♯‘(0..^4)) = 4) | |
39 | 37, 38 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^4)) = 4) |
40 | 39 | oveq2d 6830 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^4))) = (1 / 4)) |
41 | 36, 40 | oveq12d 6832 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) = ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4))) |
42 | ringmnd 18776 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
43 | 22, 42 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
44 | cnfldadd 19973 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
45 | 31, 44 | gsumws4 39020 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)))) → (ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 + (𝐵 + (𝐶 + 𝐷)))) |
46 | 43, 30, 45 | syl2anc 696 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 + (𝐵 + (𝐶 + 𝐷)))) |
47 | 46, 39 | oveq12d 6832 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4))) = ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) |
48 | 21, 41, 47 | 3brtr3d 4835 | 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 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∅c0 4058 class class class wbr 4804 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 Fincfn 8123 ℂcc 10146 0cc0 10148 1c1 10149 + caddc 10151 · cmul 10153 ≤ cle 10287 / cdiv 10896 ℕcn 11232 4c4 11284 ℕ0cn0 11504 ℝ+crp 12045 ..^cfzo 12679 ♯chash 13331 Word cword 13497 〈“cs4 13808 Σg cgsu 16323 Mndcmnd 17515 mulGrpcmgp 18709 Ringcrg 18767 ℂfldccnfld 19968 ↑𝑐ccxp 24522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-tpos 7522 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-fi 8484 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-ioo 12392 df-ioc 12393 df-ico 12394 df-icc 12395 df-fz 12540 df-fzo 12680 df-fl 12807 df-mod 12883 df-seq 13016 df-exp 13075 df-fac 13275 df-bc 13304 df-hash 13332 df-word 13505 df-concat 13507 df-s1 13508 df-s2 13813 df-s3 13814 df-s4 13815 df-shft 14026 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-limsup 14421 df-clim 14438 df-rlim 14439 df-sum 14636 df-ef 15017 df-sin 15019 df-cos 15020 df-pi 15022 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-hom 16188 df-cco 16189 df-rest 16305 df-topn 16306 df-0g 16324 df-gsum 16325 df-topgen 16326 df-pt 16327 df-prds 16330 df-xrs 16384 df-qtop 16389 df-imas 16390 df-xps 16392 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-mhm 17556 df-submnd 17557 df-grp 17646 df-minusg 17647 df-mulg 17762 df-subg 17812 df-ghm 17879 df-gim 17922 df-cntz 17970 df-cmn 18415 df-abl 18416 df-mgp 18710 df-ur 18722 df-ring 18769 df-cring 18770 df-oppr 18843 df-dvdsr 18861 df-unit 18862 df-invr 18892 df-dvr 18903 df-drng 18971 df-subrg 19000 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-fbas 19965 df-fg 19966 df-cnfld 19969 df-refld 20173 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-cld 21045 df-ntr 21046 df-cls 21047 df-nei 21124 df-lp 21162 df-perf 21163 df-cn 21253 df-cnp 21254 df-haus 21341 df-cmp 21412 df-tx 21587 df-hmeo 21780 df-fil 21871 df-fm 21963 df-flim 21964 df-flf 21965 df-xms 22346 df-ms 22347 df-tms 22348 df-cncf 22902 df-limc 23849 df-dv 23850 df-log 24523 df-cxp 24524 |
This theorem is referenced by: (None) |
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