| 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 2737 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
| 2 | fzofi 14015 | . . . 4 ⊢ (0..^4) ∈ Fin | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^4) ∈ Fin) |
| 4 | 4nn 12349 | . . . . 5 ⊢ 4 ∈ ℕ | |
| 5 | lbfzo0 13739 | . . . . 5 ⊢ (0 ∈ (0..^4) ↔ 4 ∈ ℕ) | |
| 6 | 4, 5 | mpbir 231 | . . . 4 ⊢ 0 ∈ (0..^4) |
| 7 | ne0i 4341 | . . . 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 14912 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+) |
| 14 | wrdf 14557 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶𝐷”〉 ∈ Word ℝ+ → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) | |
| 15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉:(0..^(♯‘〈“𝐴𝐵𝐶𝐷”〉))⟶ℝ+) |
| 16 | s4len 14938 | . . . . . . 7 ⊢ (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4 | |
| 17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (♯‘〈“𝐴𝐵𝐶𝐷”〉) = 4) |
| 18 | 17 | oveq2d 7447 | . . . . 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 27033 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) ≤ ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4)))) |
| 22 | cnring 21403 | . . . . 5 ⊢ ℂfld ∈ Ring | |
| 23 | 1 | ringmgp 20236 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
| 24 | 22, 23 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
| 25 | 9 | rpcnd 13079 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 26 | 10 | rpcnd 13079 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 27 | 11 | rpcnd 13079 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 28 | 12 | rpcnd 13079 | . . . . . 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 21368 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 32 | 1, 31 | mgpbas 20142 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 33 | cnfldmul 21372 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 34 | 1, 33 | mgpplusg 20141 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
| 35 | 32, 34 | gsumws4 44210 | . . . 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 12545 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
| 38 | hashfzo0 14469 | . . . . 5 ⊢ (4 ∈ ℕ0 → (♯‘(0..^4)) = 4) | |
| 39 | 37, 38 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^4)) = 4) |
| 40 | 39 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^4))) = (1 / 4)) |
| 41 | 36, 40 | oveq12d 7449 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵𝐶𝐷”〉)↑𝑐(1 / (♯‘(0..^4)))) = ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4))) |
| 42 | ringmnd 20240 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
| 43 | 22, 42 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
| 44 | cnfldadd 21370 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
| 45 | 31, 44 | gsumws4 44210 | . . . 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 7449 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵𝐶𝐷”〉) / (♯‘(0..^4))) = ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) |
| 48 | 21, 41, 47 | 3brtr3d 5174 | 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 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 / cdiv 11920 ℕcn 12266 4c4 12323 ℕ0cn0 12526 ℝ+crp 13034 ..^cfzo 13694 ♯chash 14369 Word cword 14552 〈“cs4 14882 Σg cgsu 17485 Mndcmnd 18747 mulGrpcmgp 20137 Ringcrg 20230 ℂfldccnfld 21364 ↑𝑐ccxp 26597 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-s4 14889 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-ghm 19231 df-gim 19277 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-subrng 20546 df-subrg 20570 df-drng 20731 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-refld 21623 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 |
| This theorem is referenced by: (None) |
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