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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 44194). (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 2730 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
| 2 | fzofi 13946 | . . . 4 ⊢ (0..^2) ∈ Fin | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
| 4 | 2nn 12266 | . . . . 5 ⊢ 2 ∈ ℕ | |
| 5 | lbfzo0 13667 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
| 6 | 4, 5 | mpbir 231 | . . . 4 ⊢ 0 ∈ (0..^2) |
| 7 | ne0i 4307 | . . . 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 14844 | . . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ+) |
| 12 | wrdf 14490 | . . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ+ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) | |
| 13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) |
| 14 | s2len 14862 | . . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2 | |
| 15 | 14 | oveq2i 7401 | . . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2) |
| 16 | 15 | feq2i 6683 | . . . 4 ⊢ (⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+ ↔ ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ+) |
| 17 | 13, 16 | sylib 218 | . . 3 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ+) |
| 18 | amgmw2d.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ+) | |
| 19 | amgmw2d.3 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ+) | |
| 20 | 18, 19 | s2cld 14844 | . . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ+) |
| 21 | wrdf 14490 | . . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ+ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) | |
| 22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) |
| 23 | s2len 14862 | . . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2 | |
| 24 | 23 | oveq2i 7401 | . . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2) |
| 25 | 24 | feq2i 6683 | . . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 26 | 22, 25 | sylib 218 | . . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 27 | cnring 21309 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
| 28 | ringmnd 20159 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
| 29 | 27, 28 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) |
| 30 | 18 | rpcnd 13004 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 31 | 19 | rpcnd 13004 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
| 32 | cnfldbas 21275 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 33 | cnfldadd 21277 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
| 34 | 32, 33 | gsumws2 18776 | . . . . 5 ⊢ ((ℂfld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂfld Σg ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄)) |
| 35 | 29, 30, 31, 34 | syl3anc 1373 | . . . 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 49795 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) ≤ (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩))) |
| 39 | 9, 10 | jca 511 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)) |
| 40 | 18, 19 | jca 511 | . . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ+ ∧ 𝑄 ∈ ℝ+)) |
| 41 | ofs2 14944 | . . . . 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 7406 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂfld) Σg ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩)) |
| 44 | 1 | ringmgp 20155 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
| 45 | 27, 44 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
| 46 | 18 | rpred 13002 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 47 | 9, 46 | rpcxpcld 26649 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) |
| 48 | 47 | rpcnd 13004 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) |
| 49 | 19 | rpred 13002 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
| 50 | 10, 49 | rpcxpcld 26649 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) |
| 51 | 50 | rpcnd 13004 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) |
| 52 | 1, 32 | mgpbas 20061 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 53 | cnfldmul 21279 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 54 | 1, 53 | mgpplusg 20060 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
| 55 | 52, 54 | gsumws2 18776 | . . . 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 1373 | . . 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 14944 | . . . . 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 7406 | . . 3 ⊢ (𝜑 → (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩)) = (ℂfld Σg ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)) |
| 61 | 9, 18 | rpmulcld 13018 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) |
| 62 | 61 | rpcnd 13004 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) |
| 63 | 10, 19 | rpmulcld 13018 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) |
| 64 | 63 | rpcnd 13004 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) |
| 65 | 32, 33 | gsumws2 18776 | . . . 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 1373 | . . 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 5141 | 1 ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∅c0 4299 class class class wbr 5110 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∘f cof 7654 Fincfn 8921 ℂcc 11073 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 ≤ cle 11216 ℕcn 12193 2c2 12248 ℝ+crp 12958 ..^cfzo 13622 ♯chash 14302 Word cword 14485 ⟨“cs2 14814 Σg cgsu 17410 Mndcmnd 18668 mulGrpcmgp 20056 Ringcrg 20149 ℂfldccnfld 21271 ↑𝑐ccxp 26471 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568 df-s2 14821 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-mulg 19007 df-subg 19062 df-ghm 19152 df-gim 19198 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-subrng 20462 df-subrg 20486 df-drng 20647 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-refld 21521 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775 df-log 26472 df-cxp 26473 |
| This theorem is referenced by: young2d 49798 |
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