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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 40544). (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 2821 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13336 | . . . 4 ⊢ (0..^2) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
4 | 2nn 11704 | . . . . 5 ⊢ 2 ∈ ℕ | |
5 | lbfzo0 13071 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
6 | 4, 5 | mpbir 233 | . . . 4 ⊢ 0 ∈ (0..^2) |
7 | ne0i 4300 | . . . 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 14227 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word ℝ+) |
12 | wrdf 13860 | . . . . 5 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) |
14 | s2len 14245 | . . . . . 6 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
15 | 14 | oveq2i 7161 | . . . . 5 ⊢ (0..^(♯‘〈“𝐴𝐵”〉)) = (0..^2) |
16 | 15 | feq2i 6501 | . . . 4 ⊢ (〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+ ↔ 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
17 | 13, 16 | sylib 220 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
18 | amgmw2d.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ+) | |
19 | amgmw2d.3 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ+) | |
20 | 18, 19 | s2cld 14227 | . . . . 5 ⊢ (𝜑 → 〈“𝑃𝑄”〉 ∈ Word ℝ+) |
21 | wrdf 13860 | . . . . 5 ⊢ (〈“𝑃𝑄”〉 ∈ Word ℝ+ → 〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+) |
23 | s2len 14245 | . . . . . 6 ⊢ (♯‘〈“𝑃𝑄”〉) = 2 | |
24 | 23 | oveq2i 7161 | . . . . 5 ⊢ (0..^(♯‘〈“𝑃𝑄”〉)) = (0..^2) |
25 | 24 | feq2i 6501 | . . . 4 ⊢ (〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+ ↔ 〈“𝑃𝑄”〉:(0..^2)⟶ℝ+) |
26 | 22, 25 | sylib 220 | . . 3 ⊢ (𝜑 → 〈“𝑃𝑄”〉:(0..^2)⟶ℝ+) |
27 | cnring 20561 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
28 | ringmnd 19300 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
29 | 27, 28 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) |
30 | 18 | rpcnd 12427 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
31 | 19 | rpcnd 12427 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
32 | cnfldbas 20543 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
33 | cnfldadd 20544 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
34 | 32, 33 | gsumws2 18001 | . . . . 5 ⊢ ((ℂfld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂfld Σg 〈“𝑃𝑄”〉) = (𝑃 + 𝑄)) |
35 | 29, 30, 31, 34 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (ℂfld Σg 〈“𝑃𝑄”〉) = (𝑃 + 𝑄)) |
36 | amgmw2d.4 | . . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1) | |
37 | 35, 36 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝑃𝑄”〉) = 1) |
38 | 1, 3, 8, 17, 26, 37 | amgmwlem 44896 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) ≤ (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉))) |
39 | 9, 10 | jca 514 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)) |
40 | 18, 19 | jca 514 | . . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) |
41 | ofs2 14325 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) → (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉) = 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) | |
42 | 39, 40, 41 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉) = 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) |
43 | 42 | oveq2d 7166 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) = ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉)) |
44 | 1 | ringmgp 19297 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
45 | 27, 44 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
46 | 18 | rpred 12425 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
47 | 9, 46 | rpcxpcld 25309 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) |
48 | 47 | rpcnd 12427 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) |
49 | 19 | rpred 12425 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
50 | 10, 49 | rpcxpcld 25309 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) |
51 | 50 | rpcnd 12427 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) |
52 | 1, 32 | mgpbas 19239 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
53 | cnfldmul 20545 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
54 | 1, 53 | mgpplusg 19237 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
55 | 52, 54 | gsumws2 18001 | . . . 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 1367 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
57 | 43, 56 | eqtrd 2856 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
58 | ofs2 14325 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) → (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉) = 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) | |
59 | 39, 40, 58 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉) = 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) |
60 | 59 | oveq2d 7166 | . . 3 ⊢ (𝜑 → (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉)) = (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉)) |
61 | 9, 18 | rpmulcld 12441 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) |
62 | 61 | rpcnd 12427 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) |
63 | 10, 19 | rpmulcld 12441 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) |
64 | 63 | rpcnd 12427 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) |
65 | 32, 33 | gsumws2 18001 | . . . 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 1367 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
67 | 60, 66 | eqtrd 2856 | . 2 ⊢ (𝜑 → (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
68 | 38, 57, 67 | 3brtr3d 5090 | 1 ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4291 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ∘f cof 7401 Fincfn 8503 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 ≤ cle 10670 ℕcn 11632 2c2 11686 ℝ+crp 12383 ..^cfzo 13027 ♯chash 13684 Word cword 13855 〈“cs2 14197 Σg cgsu 16708 Mndcmnd 17905 mulGrpcmgp 19233 Ringcrg 19291 ℂfldccnfld 20539 ↑𝑐ccxp 25133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-mulg 18219 df-subg 18270 df-ghm 18350 df-gim 18393 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-subrg 19527 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-refld 20743 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-cxp 25135 |
This theorem is referenced by: young2d 44899 |
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