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 2821 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13343 | . . . 4 ⊢ (0..^4) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^4) ∈ Fin) |
4 | 4nn 11721 | . . . . 5 ⊢ 4 ∈ ℕ | |
5 | lbfzo0 13078 | . . . . 5 ⊢ (0 ∈ (0..^4) ↔ 4 ∈ ℕ) | |
6 | 4, 5 | mpbir 233 | . . . 4 ⊢ 0 ∈ (0..^4) |
7 | ne0i 4300 | . . . 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 14235 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+) |
14 | wrdf 13867 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) |
16 | s4len 14261 | . . . . . . 7 ⊢ (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4) |
18 | 17 | oveq2d 7172 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉)) = (0..^4)) |
19 | 18 | feq2d 6500 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+ ↔ 〈“𝐴𝐵𝐶𝐷”〉:(0..^4)⟶ℝ+)) |
20 | 15, 19 | mpbid 234 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉:(0..^4)⟶ℝ+) |
21 | 1, 3, 8, 20 | amgmlem 25567 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) ≤ ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4)))) |
22 | cnring 20567 | . . . . 5 ⊢ ℂfld ∈ Ring | |
23 | 1 | ringmgp 19303 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
24 | 22, 23 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
25 | 9 | rpcnd 12434 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
26 | 10 | rpcnd 12434 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
27 | 11 | rpcnd 12434 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
28 | 12 | rpcnd 12434 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
29 | 27, 28 | jca 514 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) |
30 | 25, 26, 29 | jca32 518 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)))) |
31 | cnfldbas 20549 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
32 | 1, 31 | mgpbas 19245 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
33 | cnfldmul 20551 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
34 | 1, 33 | mgpplusg 19243 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
35 | 32, 34 | gsumws4 40570 | . . . 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 586 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 · (𝐵 · (𝐶 · 𝐷)))) |
37 | 4nn0 11917 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
38 | hashfzo0 13792 | . . . . 5 ⊢ (4 ∈ ℕ0 → (♯‘(0..^4)) = 4) | |
39 | 37, 38 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^4)) = 4) |
40 | 39 | oveq2d 7172 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^4))) = (1 / 4)) |
41 | 36, 40 | oveq12d 7174 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) = ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4))) |
42 | ringmnd 19306 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
43 | 22, 42 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
44 | cnfldadd 20550 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
45 | 31, 44 | gsumws4 40570 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)))) → (ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 + (𝐵 + (𝐶 + 𝐷)))) |
46 | 43, 30, 45 | syl2anc 586 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) = (𝐴 + (𝐵 + (𝐶 + 𝐷)))) |
47 | 46, 39 | oveq12d 7174 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4))) = ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) |
48 | 21, 41, 47 | 3brtr3d 5097 | 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ≤ cle 10676 / cdiv 11297 ℕcn 11638 4c4 11695 ℕ0cn0 11898 ℝ+crp 12390 ..^cfzo 13034 ♯chash 13691 Word cword 13862 〈“cs4 14205 Σg cgsu 16714 Mndcmnd 17911 mulGrpcmgp 19239 Ringcrg 19297 ℂfldccnfld 20545 ↑𝑐ccxp 25139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-s4 14212 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-mulg 18225 df-subg 18276 df-ghm 18356 df-gim 18399 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-subrg 19533 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-refld 20749 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 df-cxp 25141 |
This theorem is referenced by: (None) |
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