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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 44435). (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 2736 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
| 2 | fzofi 13897 | . . . 4 ⊢ (0..^2) ∈ Fin | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
| 4 | 2nn 12218 | . . . . 5 ⊢ 2 ∈ ℕ | |
| 5 | lbfzo0 13615 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
| 6 | 4, 5 | mpbir 231 | . . . 4 ⊢ 0 ∈ (0..^2) |
| 7 | ne0i 4293 | . . . 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 14794 | . . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ+) |
| 12 | wrdf 14441 | . . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ+ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) | |
| 13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) |
| 14 | s2len 14812 | . . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2 | |
| 15 | 14 | oveq2i 7369 | . . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2) |
| 16 | 15 | feq2i 6654 | . . . 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 14794 | . . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ+) |
| 21 | wrdf 14441 | . . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ+ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) | |
| 22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) |
| 23 | s2len 14812 | . . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2 | |
| 24 | 23 | oveq2i 7369 | . . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2) |
| 25 | 24 | feq2i 6654 | . . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 26 | 22, 25 | sylib 218 | . . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 27 | cnring 21345 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
| 28 | ringmnd 20178 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
| 29 | 27, 28 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) |
| 30 | 18 | rpcnd 12951 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 31 | 19 | rpcnd 12951 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
| 32 | cnfldbas 21313 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 33 | cnfldadd 21315 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
| 34 | 32, 33 | gsumws2 18767 | . . . . 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 2771 | . . 3 ⊢ (𝜑 → (ℂfld Σg ⟨“𝑃𝑄”⟩) = 1) |
| 38 | 1, 3, 8, 17, 26, 37 | amgmwlem 50043 | . 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 14894 | . . . . 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 7374 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂfld) Σg ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩)) |
| 44 | 1 | ringmgp 20174 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
| 45 | 27, 44 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
| 46 | 18 | rpred 12949 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 47 | 9, 46 | rpcxpcld 26698 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) |
| 48 | 47 | rpcnd 12951 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) |
| 49 | 19 | rpred 12949 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
| 50 | 10, 49 | rpcxpcld 26698 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) |
| 51 | 50 | rpcnd 12951 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) |
| 52 | 1, 32 | mgpbas 20080 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 53 | cnfldmul 21317 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 54 | 1, 53 | mgpplusg 20079 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
| 55 | 52, 54 | gsumws2 18767 | . . . 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 2771 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) |
| 58 | ofs2 14894 | . . . . 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 7374 | . . 3 ⊢ (𝜑 → (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩)) = (ℂfld Σg ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)) |
| 61 | 9, 18 | rpmulcld 12965 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) |
| 62 | 61 | rpcnd 12951 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) |
| 63 | 10, 19 | rpmulcld 12965 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) |
| 64 | 63 | rpcnd 12951 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) |
| 65 | 32, 33 | gsumws2 18767 | . . . 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 2771 | . 2 ⊢ (𝜑 → (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| 68 | 38, 57, 67 | 3brtr3d 5129 | 1 ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∅c0 4285 class class class wbr 5098 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∘f cof 7620 Fincfn 8883 ℂcc 11024 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 ≤ cle 11167 ℕcn 12145 2c2 12200 ℝ+crp 12905 ..^cfzo 13570 ♯chash 14253 Word cword 14436 ⟨“cs2 14764 Σg cgsu 17360 Mndcmnd 18659 mulGrpcmgp 20075 Ringcrg 20168 ℂfldccnfld 21309 ↑𝑐ccxp 26520 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-word 14437 df-concat 14494 df-s1 14520 df-s2 14771 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-mulg 18998 df-subg 19053 df-ghm 19142 df-gim 19188 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-subrng 20479 df-subrg 20503 df-drng 20664 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-refld 21560 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-log 26521 df-cxp 26522 |
| This theorem is referenced by: young2d 50046 |
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