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Mirrors > Home > MPE Home > Th. List > Mathboxes > amgmw2d | Structured version Visualization version GIF version |
Description: Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 40429). (Contributed by Kunhao Zheng, 20-Jun-2021.) |
Ref | Expression |
---|---|
amgmw2d.0 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
amgmw2d.1 | ⊢ (𝜑 → 𝑃 ∈ ℝ+) |
amgmw2d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
amgmw2d.3 | ⊢ (𝜑 → 𝑄 ∈ ℝ+) |
amgmw2d.4 | ⊢ (𝜑 → (𝑃 + 𝑄) = 1) |
Ref | Expression |
---|---|
amgmw2d | ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13330 | . . . 4 ⊢ (0..^2) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
4 | 2nn 11698 | . . . . 5 ⊢ 2 ∈ ℕ | |
5 | lbfzo0 13065 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
6 | 4, 5 | mpbir 232 | . . . 4 ⊢ 0 ∈ (0..^2) |
7 | ne0i 4297 | . . . 4 ⊢ (0 ∈ (0..^2) → (0..^2) ≠ ∅) | |
8 | 6, 7 | mp1i 13 | . . 3 ⊢ (𝜑 → (0..^2) ≠ ∅) |
9 | amgmw2d.0 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | amgmw2d.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
11 | 9, 10 | s2cld 14221 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word ℝ+) |
12 | wrdf 13854 | . . . . 5 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) |
14 | s2len 14239 | . . . . . 6 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
15 | 14 | oveq2i 7156 | . . . . 5 ⊢ (0..^(♯‘〈“𝐴𝐵”〉)) = (0..^2) |
16 | 15 | feq2i 6499 | . . . 4 ⊢ (〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+ ↔ 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
17 | 13, 16 | sylib 219 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
18 | amgmw2d.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ+) | |
19 | amgmw2d.3 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ+) | |
20 | 18, 19 | s2cld 14221 | . . . . 5 ⊢ (𝜑 → 〈“𝑃𝑄”〉 ∈ Word ℝ+) |
21 | wrdf 13854 | . . . . 5 ⊢ (〈“𝑃𝑄”〉 ∈ Word ℝ+ → 〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+) |
23 | s2len 14239 | . . . . . 6 ⊢ (♯‘〈“𝑃𝑄”〉) = 2 | |
24 | 23 | oveq2i 7156 | . . . . 5 ⊢ (0..^(♯‘〈“𝑃𝑄”〉)) = (0..^2) |
25 | 24 | feq2i 6499 | . . . 4 ⊢ (〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+ ↔ 〈“𝑃𝑄”〉:(0..^2)⟶ℝ+) |
26 | 22, 25 | sylib 219 | . . 3 ⊢ (𝜑 → 〈“𝑃𝑄”〉:(0..^2)⟶ℝ+) |
27 | cnring 20495 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
28 | ringmnd 19235 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
29 | 27, 28 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) |
30 | 18 | rpcnd 12421 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
31 | 19 | rpcnd 12421 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
32 | cnfldbas 20477 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
33 | cnfldadd 20478 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
34 | 32, 33 | gsumws2 17995 | . . . . 5 ⊢ ((ℂfld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂfld Σg 〈“𝑃𝑄”〉) = (𝑃 + 𝑄)) |
35 | 29, 30, 31, 34 | syl3anc 1363 | . . . 4 ⊢ (𝜑 → (ℂfld Σg 〈“𝑃𝑄”〉) = (𝑃 + 𝑄)) |
36 | amgmw2d.4 | . . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1) | |
37 | 35, 36 | eqtrd 2853 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝑃𝑄”〉) = 1) |
38 | 1, 3, 8, 17, 26, 37 | amgmwlem 44831 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) ≤ (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉))) |
39 | 9, 10 | jca 512 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)) |
40 | 18, 19 | jca 512 | . . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) |
41 | ofs2 14319 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) → (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉) = 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) | |
42 | 39, 40, 41 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉) = 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) |
43 | 42 | oveq2d 7161 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) = ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉)) |
44 | 1 | ringmgp 19232 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
45 | 27, 44 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
46 | 18 | rpred 12419 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
47 | 9, 46 | rpcxpcld 25242 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) |
48 | 47 | rpcnd 12421 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) |
49 | 19 | rpred 12419 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
50 | 10, 49 | rpcxpcld 25242 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) |
51 | 50 | rpcnd 12421 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) |
52 | 1, 32 | mgpbas 19174 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
53 | cnfldmul 20479 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
54 | 1, 53 | mgpplusg 19172 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
55 | 52, 54 | gsumws2 17995 | . . . 4 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ (𝐴↑𝑐𝑃) ∈ ℂ ∧ (𝐵↑𝑐𝑄) ∈ ℂ) → ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
56 | 45, 48, 51, 55 | syl3anc 1363 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
57 | 43, 56 | eqtrd 2853 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
58 | ofs2 14319 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) → (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉) = 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) | |
59 | 39, 40, 58 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉) = 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) |
60 | 59 | oveq2d 7161 | . . 3 ⊢ (𝜑 → (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉)) = (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉)) |
61 | 9, 18 | rpmulcld 12435 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) |
62 | 61 | rpcnd 12421 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) |
63 | 10, 19 | rpmulcld 12435 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) |
64 | 63 | rpcnd 12421 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) |
65 | 32, 33 | gsumws2 17995 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ (𝐴 · 𝑃) ∈ ℂ ∧ (𝐵 · 𝑄) ∈ ℂ) → (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
66 | 29, 62, 64, 65 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
67 | 60, 66 | eqtrd 2853 | . 2 ⊢ (𝜑 → (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
68 | 38, 57, 67 | 3brtr3d 5088 | 1 ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Fincfn 8497 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 ≤ cle 10664 ℕcn 11626 2c2 11680 ℝ+crp 12377 ..^cfzo 13021 ♯chash 13678 Word cword 13849 〈“cs2 14191 Σg cgsu 16702 Mndcmnd 17899 mulGrpcmgp 19168 Ringcrg 19226 ℂfldccnfld 20473 ↑𝑐ccxp 25066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-mulg 18163 df-subg 18214 df-ghm 18294 df-gim 18337 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-subrg 19462 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-refld 20677 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-cmp 21923 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-cxp 25068 |
This theorem is referenced by: young2d 44834 |
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