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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 44188). (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 2735 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
| 2 | fzofi 14012 | . . . 4 ⊢ (0..^2) ∈ Fin | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
| 4 | 2nn 12337 | . . . . 5 ⊢ 2 ∈ ℕ | |
| 5 | lbfzo0 13736 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
| 6 | 4, 5 | mpbir 231 | . . . 4 ⊢ 0 ∈ (0..^2) |
| 7 | ne0i 4347 | . . . 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 14907 | . . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ+) |
| 12 | wrdf 14554 | . . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ+ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) | |
| 13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) |
| 14 | s2len 14925 | . . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2 | |
| 15 | 14 | oveq2i 7442 | . . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2) |
| 16 | 15 | feq2i 6729 | . . . 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 14907 | . . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ+) |
| 21 | wrdf 14554 | . . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ+ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) | |
| 22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) |
| 23 | s2len 14925 | . . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2 | |
| 24 | 23 | oveq2i 7442 | . . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2) |
| 25 | 24 | feq2i 6729 | . . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 26 | 22, 25 | sylib 218 | . . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 27 | cnring 21421 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
| 28 | ringmnd 20261 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
| 29 | 27, 28 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) |
| 30 | 18 | rpcnd 13077 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 31 | 19 | rpcnd 13077 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
| 32 | cnfldbas 21386 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 33 | cnfldadd 21388 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
| 34 | 32, 33 | gsumws2 18868 | . . . . 5 ⊢ ((ℂfld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂfld Σg ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄)) |
| 35 | 29, 30, 31, 34 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (ℂfld Σg ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄)) |
| 36 | amgmw2d.4 | . . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1) | |
| 37 | 35, 36 | eqtrd 2775 | . . 3 ⊢ (𝜑 → (ℂfld Σg ⟨“𝑃𝑄”⟩) = 1) |
| 38 | 1, 3, 8, 17, 26, 37 | amgmwlem 49033 | . 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 15007 | . . . . 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 7447 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂfld) Σg ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩)) |
| 44 | 1 | ringmgp 20257 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
| 45 | 27, 44 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
| 46 | 18 | rpred 13075 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 47 | 9, 46 | rpcxpcld 26790 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) |
| 48 | 47 | rpcnd 13077 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) |
| 49 | 19 | rpred 13075 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
| 50 | 10, 49 | rpcxpcld 26790 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) |
| 51 | 50 | rpcnd 13077 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) |
| 52 | 1, 32 | mgpbas 20158 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 53 | cnfldmul 21390 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 54 | 1, 53 | mgpplusg 20156 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
| 55 | 52, 54 | gsumws2 18868 | . . . 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 1370 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
| 57 | 43, 56 | eqtrd 2775 | . 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 584 | . . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) |
| 60 | 59 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩)) = (ℂfld Σg ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)) |
| 61 | 9, 18 | rpmulcld 13091 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) |
| 62 | 61 | rpcnd 13077 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) |
| 63 | 10, 19 | rpmulcld 13091 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) |
| 64 | 63 | rpcnd 13077 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) |
| 65 | 32, 33 | gsumws2 18868 | . . . 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 1370 | . . 3 ⊢ (𝜑 → (ℂfld Σg ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| 67 | 60, 66 | eqtrd 2775 | . 2 ⊢ (𝜑 → (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| 68 | 38, 57, 67 | 3brtr3d 5179 | 1 ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Fincfn 8984 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ≤ cle 11294 ℕcn 12264 2c2 12319 ℝ+crp 13032 ..^cfzo 13691 ♯chash 14366 Word cword 14549 ⟨“cs2 14877 Σg cgsu 17487 Mndcmnd 18760 mulGrpcmgp 20152 Ringcrg 20251 ℂfldccnfld 21382 ↑𝑐ccxp 26612 |
| 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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-mulg 19099 df-subg 19154 df-ghm 19244 df-gim 19290 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-subrng 20563 df-subrg 20587 df-drng 20748 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-refld 21641 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614 |
| This theorem is referenced by: young2d 49036 |
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