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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 44815). (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 2769 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
| 2 | fzofi 14009 | . . . 4 ⊢ (0..^2) ∈ Fin | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
| 4 | 2nn 12313 | . . . . 5 ⊢ 2 ∈ ℕ | |
| 5 | lbfzo0 13727 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
| 6 | 4, 5 | mpbir 234 | . . . 4 ⊢ 0 ∈ (0..^2) |
| 7 | ne0i 4302 | . . . 4 ⊢ (0 ∈ (0..^2) → (0..^2) ≠ ∅) | |
| 8 | 6, 7 | mp1i 14 | . . 3 ⊢ (𝜑 → (0..^2) ≠ ∅) |
| 9 | amgmw2d.0 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
| 10 | amgmw2d.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
| 11 | 9, 10 | s2cld 14907 | . . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ+) |
| 12 | wrdf 14554 | . . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ+ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) | |
| 13 | 11, 12 | syl 18 | . . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) |
| 14 | s2len 14925 | . . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2 | |
| 15 | 14 | oveq2i 7422 | . . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2) |
| 16 | 15 | feq2i 6698 | . . . 4 ⊢ (⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+ ↔ ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ+) |
| 17 | 13, 16 | sylib 221 | . . 3 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ+) |
| 18 | amgmw2d.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ+) | |
| 19 | amgmw2d.3 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ+) | |
| 20 | 18, 19 | s2cld 14907 | . . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ+) |
| 21 | wrdf 14554 | . . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ+ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) | |
| 22 | 20, 21 | syl 18 | . . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) |
| 23 | s2len 14925 | . . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2 | |
| 24 | 23 | oveq2i 7422 | . . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2) |
| 25 | 24 | feq2i 6698 | . . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 26 | 22, 25 | sylib 221 | . . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 27 | cnring 21512 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
| 28 | ringmnd 20324 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
| 29 | 27, 28 | mp1i 14 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) |
| 30 | 18 | rpcnd 13061 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 31 | 19 | rpcnd 13061 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
| 32 | cnfldbas 21494 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 33 | cnfldadd 21496 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
| 34 | 32, 33 | gsumws2 18900 | . . . . 5 ⊢ ((ℂfld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂfld Σg ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄)) |
| 35 | 29, 30, 31, 34 | syl3anc 1396 | . . . 4 ⊢ (𝜑 → (ℂfld Σg ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄)) |
| 36 | amgmw2d.4 | . . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1) | |
| 37 | 35, 36 | eqtrd 2804 | . . 3 ⊢ (𝜑 → (ℂfld Σg ⟨“𝑃𝑄”⟩) = 1) |
| 38 | 1, 3, 8, 17, 26, 37 | amgmwlem 50475 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) ≤ (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩))) |
| 39 | 9, 10 | jca 520 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)) |
| 40 | 18, 19 | jca 520 | . . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) |
| 41 | ofs2 15007 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) → (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩) = ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩) | |
| 42 | 39, 40, 41 | syl2anc 595 | . . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩) = ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩) |
| 43 | 42 | oveq2d 7427 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂfld) Σg ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩)) |
| 44 | 1 | ringmgp 20320 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
| 45 | 27, 44 | mp1i 14 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
| 46 | 18 | rpred 13059 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 47 | 9, 46 | rpcxpcld 26863 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) |
| 48 | 47 | rpcnd 13061 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) |
| 49 | 19 | rpred 13059 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
| 50 | 10, 49 | rpcxpcld 26863 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) |
| 51 | 50 | rpcnd 13061 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) |
| 52 | 1, 32 | mgpbas 20220 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 53 | cnfldmul 21498 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 54 | 1, 53 | mgpplusg 20219 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
| 55 | 52, 54 | gsumws2 18900 | . . . 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 1396 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
| 57 | 43, 56 | eqtrd 2804 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
| 58 | ofs2 15007 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) → (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) | |
| 59 | 39, 40, 58 | syl2anc 595 | . . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) |
| 60 | 59 | oveq2d 7427 | . . 3 ⊢ (𝜑 → (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩)) = (ℂfld Σg ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)) |
| 61 | 9, 18 | rpmulcld 13075 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) |
| 62 | 61 | rpcnd 13061 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) |
| 63 | 10, 19 | rpmulcld 13075 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) |
| 64 | 63 | rpcnd 13061 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) |
| 65 | 32, 33 | gsumws2 18900 | . . . 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 1396 | . . 3 ⊢ (𝜑 → (ℂfld Σg ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| 67 | 60, 66 | eqtrd 2804 | . 2 ⊢ (𝜑 → (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| 68 | 38, 57, 67 | 3brtr3d 5146 | 1 ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∅c0 4294 class class class wbr 5113 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ∘f cof 7673 Fincfn 8942 ℂcc 11097 0cc0 11099 1c1 11100 + caddc 11102 · cmul 11104 ≤ cle 11243 ℕcn 12232 2c2 12294 ℝ+crp 13015 ..^cfzo 13681 ♯chash 14365 Word cword 14549 ⟨“cs2 14877 Σg cgsu 17492 Mndcmnd 18791 mulGrpcmgp 20215 Ringcrg 20314 ℂfldccnfld 21490 ↑𝑐ccxp 26685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178 ax-mulf 11179 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ioc 13376 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-fac 14309 df-bc 14338 df-hash 14366 df-word 14550 df-concat 14607 df-s1 14633 df-s2 14884 df-shft 15103 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-limsup 15521 df-clim 15538 df-rlim 15539 df-sum 15737 df-ef 16120 df-sin 16122 df-cos 16123 df-pi 16125 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-mulg 19133 df-subg 19188 df-ghm 19283 df-gim 19328 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-subrng 20630 df-subrg 20654 df-drng 20814 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-fbas 21487 df-fg 21488 df-cnfld 21491 df-refld 21723 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cld 23144 df-ntr 23145 df-cls 23146 df-nei 23223 df-lp 23261 df-perf 23262 df-cn 23352 df-cnp 23353 df-haus 23440 df-cmp 23512 df-tx 23687 df-hmeo 23880 df-fil 23971 df-fm 24063 df-flim 24064 df-flf 24065 df-xms 24445 df-ms 24446 df-tms 24447 df-cncf 25005 df-limc 25993 df-dv 25994 df-log 26686 df-cxp 26687 |
| This theorem is referenced by: young2d 50478 |
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