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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 44189). (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 13997 | . . . 4 ⊢ (0..^2) ∈ Fin | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
| 4 | 2nn 12318 | . . . . 5 ⊢ 2 ∈ ℕ | |
| 5 | lbfzo0 13721 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
| 6 | 4, 5 | mpbir 231 | . . . 4 ⊢ 0 ∈ (0..^2) |
| 7 | ne0i 4321 | . . . 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 14895 | . . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ+) |
| 12 | wrdf 14541 | . . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ+ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) | |
| 13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ+) |
| 14 | s2len 14913 | . . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2 | |
| 15 | 14 | oveq2i 7421 | . . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2) |
| 16 | 15 | feq2i 6703 | . . . 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 14895 | . . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ+) |
| 21 | wrdf 14541 | . . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ+ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) | |
| 22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+) |
| 23 | s2len 14913 | . . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2 | |
| 24 | 23 | oveq2i 7421 | . . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2) |
| 25 | 24 | feq2i 6703 | . . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ+ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 26 | 22, 25 | sylib 218 | . . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ+) |
| 27 | cnring 21358 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
| 28 | ringmnd 20208 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
| 29 | 27, 28 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) |
| 30 | 18 | rpcnd 13058 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 31 | 19 | rpcnd 13058 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
| 32 | cnfldbas 21324 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 33 | cnfldadd 21326 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
| 34 | 32, 33 | gsumws2 18825 | . . . . 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 49633 | . 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 14995 | . . . . 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 7426 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (⟨“𝐴𝐵”⟩ ∘f ↑𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂfld) Σg ⟨“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”⟩)) |
| 44 | 1 | ringmgp 20204 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
| 45 | 27, 44 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
| 46 | 18 | rpred 13056 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 47 | 9, 46 | rpcxpcld 26699 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) |
| 48 | 47 | rpcnd 13058 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) |
| 49 | 19 | rpred 13056 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
| 50 | 10, 49 | rpcxpcld 26699 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) |
| 51 | 50 | rpcnd 13058 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) |
| 52 | 1, 32 | mgpbas 20110 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 53 | cnfldmul 21328 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 54 | 1, 53 | mgpplusg 20109 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
| 55 | 52, 54 | gsumws2 18825 | . . . 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 14995 | . . . . 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 7426 | . . 3 ⊢ (𝜑 → (ℂfld Σg (⟨“𝐴𝐵”⟩ ∘f · ⟨“𝑃𝑄”⟩)) = (ℂfld Σg ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)) |
| 61 | 9, 18 | rpmulcld 13072 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) |
| 62 | 61 | rpcnd 13058 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) |
| 63 | 10, 19 | rpmulcld 13072 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) |
| 64 | 63 | rpcnd 13058 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) |
| 65 | 32, 33 | gsumws2 18825 | . . . 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 5155 | 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 2933 ∅c0 4313 class class class wbr 5124 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∘f cof 7674 Fincfn 8964 ℂcc 11132 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 ≤ cle 11275 ℕcn 12245 2c2 12300 ℝ+crp 13013 ..^cfzo 13676 ♯chash 14353 Word cword 14536 ⟨“cs2 14865 Σg cgsu 17459 Mndcmnd 18717 mulGrpcmgp 20105 Ringcrg 20198 ℂfldccnfld 21320 ↑𝑐ccxp 26521 |
| 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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214 |
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-word 14537 df-concat 14594 df-s1 14619 df-s2 14872 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-pi 16093 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-mulg 19056 df-subg 19111 df-ghm 19201 df-gim 19247 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-subrng 20511 df-subrg 20535 df-drng 20696 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-refld 21570 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825 df-log 26522 df-cxp 26523 |
| This theorem is referenced by: young2d 49636 |
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