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 2823 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13345 | . . . 4 ⊢ (0..^4) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^4) ∈ Fin) |
4 | 4nn 11723 | . . . . 5 ⊢ 4 ∈ ℕ | |
5 | lbfzo0 13080 | . . . . 5 ⊢ (0 ∈ (0..^4) ↔ 4 ∈ ℕ) | |
6 | 4, 5 | mpbir 233 | . . . 4 ⊢ 0 ∈ (0..^4) |
7 | ne0i 4302 | . . . 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 14237 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+) |
14 | wrdf 13869 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) |
16 | s4len 14263 | . . . . . . 7 ⊢ (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4) |
18 | 17 | oveq2d 7174 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉)) = (0..^4)) |
19 | 18 | feq2d 6502 | . . . 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 25569 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) ≤ ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4)))) |
22 | cnring 20569 | . . . . 5 ⊢ ℂfld ∈ Ring | |
23 | 1 | ringmgp 19305 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
24 | 22, 23 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
25 | 9 | rpcnd 12436 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
26 | 10 | rpcnd 12436 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
27 | 11 | rpcnd 12436 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
28 | 12 | rpcnd 12436 | . . . . . 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 20551 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
32 | 1, 31 | mgpbas 19247 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
33 | cnfldmul 20553 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
34 | 1, 33 | mgpplusg 19245 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
35 | 32, 34 | gsumws4 40557 | . . . 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 11919 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
38 | hashfzo0 13794 | . . . . 5 ⊢ (4 ∈ ℕ0 → (♯‘(0..^4)) = 4) | |
39 | 37, 38 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^4)) = 4) |
40 | 39 | oveq2d 7174 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^4))) = (1 / 4)) |
41 | 36, 40 | oveq12d 7176 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) = ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4))) |
42 | ringmnd 19308 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
43 | 22, 42 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
44 | cnfldadd 20552 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
45 | 31, 44 | gsumws4 40557 | . . . 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 7176 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4))) = ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) |
48 | 21, 41, 47 | 3brtr3d 5099 | 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 3018 ∅c0 4293 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ≤ cle 10678 / cdiv 11299 ℕcn 11640 4c4 11697 ℕ0cn0 11900 ℝ+crp 12392 ..^cfzo 13036 ♯chash 13693 Word cword 13864 〈“cs4 14207 Σg cgsu 16716 Mndcmnd 17913 mulGrpcmgp 19241 Ringcrg 19299 ℂfldccnfld 20547 ↑𝑐ccxp 25141 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-s4 14214 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-subrg 19535 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-refld 20751 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-cxp 25143 |
This theorem is referenced by: (None) |
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