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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > amgm2d | Structured version Visualization version GIF version |
Description: Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 25249. (Contributed by Stanislas Polu, 8-Sep-2020.) |
Ref | Expression |
---|---|
amgm2d.0 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
amgm2d.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
amgm2d | ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2795 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13192 | . . . 4 ⊢ (0..^2) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
4 | 2nn 11558 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | lbfzo0 12927 | . . . . . 6 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
6 | 4, 5 | mpbir 232 | . . . . 5 ⊢ 0 ∈ (0..^2) |
7 | 6 | ne0ii 4223 | . . . 4 ⊢ (0..^2) ≠ ∅ |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ≠ ∅) |
9 | amgm2d.0 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | amgm2d.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
11 | 9, 10 | s2cld 14069 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word ℝ+) |
12 | wrdf 13712 | . . . . 5 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) | |
13 | s2len 14087 | . . . . . . . 8 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
14 | 13 | eqcomi 2804 | . . . . . . 7 ⊢ 2 = (♯‘〈“𝐴𝐵”〉) |
15 | 14 | oveq2i 7027 | . . . . . 6 ⊢ (0..^2) = (0..^(♯‘〈“𝐴𝐵”〉)) |
16 | 15 | feq2i 6374 | . . . . 5 ⊢ (〈“𝐴𝐵”〉:(0..^2)⟶ℝ+ ↔ 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) |
17 | 12, 16 | sylibr 235 | . . . 4 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
18 | 11, 17 | syl 17 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
19 | 1, 3, 8, 18 | amgmlem 25249 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉)↑𝑐(1 / (♯‘(0..^2)))) ≤ ((ℂfld Σg 〈“𝐴𝐵”〉) / (♯‘(0..^2)))) |
20 | cnring 20249 | . . . . 5 ⊢ ℂfld ∈ Ring | |
21 | 1 | ringmgp 18993 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
22 | 20, 21 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
23 | 9 | rpcnd 12283 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
24 | 10 | rpcnd 12283 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
25 | cnfldbas 20231 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
26 | 1, 25 | mgpbas 18935 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
27 | cnfldmul 20233 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
28 | 1, 27 | mgpplusg 18933 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
29 | 26, 28 | gsumws2 17818 | . . . 4 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉) = (𝐴 · 𝐵)) |
30 | 22, 23, 24, 29 | syl3anc 1364 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉) = (𝐴 · 𝐵)) |
31 | 2nn0 11762 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
32 | hashfzo0 13639 | . . . . 5 ⊢ (2 ∈ ℕ0 → (♯‘(0..^2)) = 2) | |
33 | 31, 32 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^2)) = 2) |
34 | 33 | oveq2d 7032 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^2))) = (1 / 2)) |
35 | 30, 34 | oveq12d 7034 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉)↑𝑐(1 / (♯‘(0..^2)))) = ((𝐴 · 𝐵)↑𝑐(1 / 2))) |
36 | ringmnd 18996 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
37 | 20, 36 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
38 | cnfldadd 20232 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
39 | 25, 38 | gsumws2 17818 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℂfld Σg 〈“𝐴𝐵”〉) = (𝐴 + 𝐵)) |
40 | 37, 23, 24, 39 | syl3anc 1364 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝐴𝐵”〉) = (𝐴 + 𝐵)) |
41 | 40, 33 | oveq12d 7034 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵”〉) / (♯‘(0..^2))) = ((𝐴 + 𝐵) / 2)) |
42 | 19, 35, 41 | 3brtr3d 4993 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∅c0 4211 class class class wbr 4962 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 Fincfn 8357 ℂcc 10381 0cc0 10383 1c1 10384 + caddc 10386 · cmul 10388 ≤ cle 10522 / cdiv 11145 ℕcn 11486 2c2 11540 ℕ0cn0 11745 ℝ+crp 12239 ..^cfzo 12883 ♯chash 13540 Word cword 13707 〈“cs2 14039 Σg cgsu 16543 Mndcmnd 17733 mulGrpcmgp 18929 Ringcrg 18987 ℂfldccnfld 20227 ↑𝑐ccxp 24820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-word 13708 df-concat 13769 df-s1 13794 df-s2 14046 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-sin 15256 df-cos 15257 df-pi 15259 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mhm 17774 df-submnd 17775 df-grp 17864 df-minusg 17865 df-mulg 17982 df-subg 18030 df-ghm 18097 df-gim 18140 df-cntz 18188 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-subrg 19223 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-refld 20431 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-cmp 21679 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 df-limc 24147 df-dv 24148 df-log 24821 df-cxp 24822 |
This theorem is referenced by: (None) |
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