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Mathbox for Kunhao Zheng |
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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 43767). (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 2725 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13975 | . . . 4 ⊢ (0..^2) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
4 | 2nn 12318 | . . . . 5 ⊢ 2 ∈ ℕ | |
5 | lbfzo0 13707 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
6 | 4, 5 | mpbir 230 | . . . 4 ⊢ 0 ∈ (0..^2) |
7 | ne0i 4334 | . . . 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 14858 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word ℝ+) |
12 | wrdf 14505 | . . . . 5 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) |
14 | s2len 14876 | . . . . . 6 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
15 | 14 | oveq2i 7430 | . . . . 5 ⊢ (0..^(♯‘〈“𝐴𝐵”〉)) = (0..^2) |
16 | 15 | feq2i 6715 | . . . 4 ⊢ (〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+ ↔ 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
17 | 13, 16 | sylib 217 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
18 | amgmw2d.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ+) | |
19 | amgmw2d.3 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ+) | |
20 | 18, 19 | s2cld 14858 | . . . . 5 ⊢ (𝜑 → 〈“𝑃𝑄”〉 ∈ Word ℝ+) |
21 | wrdf 14505 | . . . . 5 ⊢ (〈“𝑃𝑄”〉 ∈ Word ℝ+ → 〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+) |
23 | s2len 14876 | . . . . . 6 ⊢ (♯‘〈“𝑃𝑄”〉) = 2 | |
24 | 23 | oveq2i 7430 | . . . . 5 ⊢ (0..^(♯‘〈“𝑃𝑄”〉)) = (0..^2) |
25 | 24 | feq2i 6715 | . . . 4 ⊢ (〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+ ↔ 〈“𝑃𝑄”〉:(0..^2)⟶ℝ+) |
26 | 22, 25 | sylib 217 | . . 3 ⊢ (𝜑 → 〈“𝑃𝑄”〉:(0..^2)⟶ℝ+) |
27 | cnring 21335 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
28 | ringmnd 20195 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
29 | 27, 28 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) |
30 | 18 | rpcnd 13053 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
31 | 19 | rpcnd 13053 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
32 | cnfldbas 21300 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
33 | cnfldadd 21302 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
34 | 32, 33 | gsumws2 18802 | . . . . 5 ⊢ ((ℂfld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂfld Σg 〈“𝑃𝑄”〉) = (𝑃 + 𝑄)) |
35 | 29, 30, 31, 34 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (ℂfld Σg 〈“𝑃𝑄”〉) = (𝑃 + 𝑄)) |
36 | amgmw2d.4 | . . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1) | |
37 | 35, 36 | eqtrd 2765 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝑃𝑄”〉) = 1) |
38 | 1, 3, 8, 17, 26, 37 | amgmwlem 48418 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) ≤ (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉))) |
39 | 9, 10 | jca 510 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)) |
40 | 18, 19 | jca 510 | . . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) |
41 | ofs2 14954 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) → (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉) = 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) | |
42 | 39, 40, 41 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉) = 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) |
43 | 42 | oveq2d 7435 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) = ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉)) |
44 | 1 | ringmgp 20191 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
45 | 27, 44 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
46 | 18 | rpred 13051 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
47 | 9, 46 | rpcxpcld 26712 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) |
48 | 47 | rpcnd 13053 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) |
49 | 19 | rpred 13051 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
50 | 10, 49 | rpcxpcld 26712 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) |
51 | 50 | rpcnd 13053 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) |
52 | 1, 32 | mgpbas 20092 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
53 | cnfldmul 21304 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
54 | 1, 53 | mgpplusg 20090 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
55 | 52, 54 | gsumws2 18802 | . . . 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 1368 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
57 | 43, 56 | eqtrd 2765 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
58 | ofs2 14954 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) → (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉) = 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) | |
59 | 39, 40, 58 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉) = 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) |
60 | 59 | oveq2d 7435 | . . 3 ⊢ (𝜑 → (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉)) = (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉)) |
61 | 9, 18 | rpmulcld 13067 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) |
62 | 61 | rpcnd 13053 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) |
63 | 10, 19 | rpmulcld 13067 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) |
64 | 63 | rpcnd 13053 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) |
65 | 32, 33 | gsumws2 18802 | . . . 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 1368 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
67 | 60, 66 | eqtrd 2765 | . 2 ⊢ (𝜑 → (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
68 | 38, 57, 67 | 3brtr3d 5180 | 1 ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∅c0 4322 class class class wbr 5149 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 Fincfn 8964 ℂcc 11138 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 ≤ cle 11281 ℕcn 12245 2c2 12300 ℝ+crp 13009 ..^cfzo 13662 ♯chash 14325 Word cword 14500 〈“cs2 14828 Σg cgsu 17425 Mndcmnd 18697 mulGrpcmgp 20086 Ringcrg 20185 ℂfldccnfld 21296 ↑𝑐ccxp 26534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-fac 14269 df-bc 14298 df-hash 14326 df-word 14501 df-concat 14557 df-s1 14582 df-s2 14835 df-shft 15050 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-sum 15669 df-ef 16047 df-sin 16049 df-cos 16050 df-pi 16052 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-mulg 19032 df-subg 19086 df-ghm 19176 df-gim 19222 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-subrng 20495 df-subrg 20520 df-drng 20638 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-refld 21554 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-limc 25839 df-dv 25840 df-log 26535 df-cxp 26536 |
This theorem is referenced by: young2d 48421 |
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