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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 44216). (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 14016 | . . . 4 ⊢ (0..^2) ∈ Fin | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) | 
| 4 | 2nn 12340 | . . . . 5 ⊢ 2 ∈ ℕ | |
| 5 | lbfzo0 13740 | . . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
| 6 | 4, 5 | mpbir 231 | . . . 4 ⊢ 0 ∈ (0..^2) | 
| 7 | ne0i 4340 | . . . 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 14911 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word ℝ+) | 
| 12 | wrdf 14558 | . . . . 5 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) | |
| 13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) | 
| 14 | s2len 14929 | . . . . . 6 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
| 15 | 14 | oveq2i 7443 | . . . . 5 ⊢ (0..^(♯‘〈“𝐴𝐵”〉)) = (0..^2) | 
| 16 | 15 | feq2i 6727 | . . . 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 14911 | . . . . 5 ⊢ (𝜑 → 〈“𝑃𝑄”〉 ∈ Word ℝ+) | 
| 21 | wrdf 14558 | . . . . 5 ⊢ (〈“𝑃𝑄”〉 ∈ Word ℝ+ → 〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+) | |
| 22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → 〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+) | 
| 23 | s2len 14929 | . . . . . 6 ⊢ (♯‘〈“𝑃𝑄”〉) = 2 | |
| 24 | 23 | oveq2i 7443 | . . . . 5 ⊢ (0..^(♯‘〈“𝑃𝑄”〉)) = (0..^2) | 
| 25 | 24 | feq2i 6727 | . . . 4 ⊢ (〈“𝑃𝑄”〉:(0..^(♯‘〈“𝑃𝑄”〉))⟶ℝ+ ↔ 〈“𝑃𝑄”〉:(0..^2)⟶ℝ+) | 
| 26 | 22, 25 | sylib 218 | . . 3 ⊢ (𝜑 → 〈“𝑃𝑄”〉:(0..^2)⟶ℝ+) | 
| 27 | cnring 21404 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
| 28 | ringmnd 20241 | . . . . . 6 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
| 29 | 27, 28 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ℂfld ∈ Mnd) | 
| 30 | 18 | rpcnd 13080 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) | 
| 31 | 19 | rpcnd 13080 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ) | 
| 32 | cnfldbas 21369 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 33 | cnfldadd 21371 | . . . . . 6 ⊢ + = (+g‘ℂfld) | |
| 34 | 32, 33 | gsumws2 18856 | . . . . 5 ⊢ ((ℂfld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂfld Σg 〈“𝑃𝑄”〉) = (𝑃 + 𝑄)) | 
| 35 | 29, 30, 31, 34 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (ℂfld Σg 〈“𝑃𝑄”〉) = (𝑃 + 𝑄)) | 
| 36 | amgmw2d.4 | . . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1) | |
| 37 | 35, 36 | eqtrd 2776 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝑃𝑄”〉) = 1) | 
| 38 | 1, 3, 8, 17, 26, 37 | amgmwlem 49376 | . 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 15011 | . . . . 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 7448 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) = ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉)) | 
| 44 | 1 | ringmgp 20237 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) | 
| 45 | 27, 44 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) | 
| 46 | 18 | rpred 13078 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ) | 
| 47 | 9, 46 | rpcxpcld 26776 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℝ+) | 
| 48 | 47 | rpcnd 13080 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑐𝑃) ∈ ℂ) | 
| 49 | 19 | rpred 13078 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ) | 
| 50 | 10, 49 | rpcxpcld 26776 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℝ+) | 
| 51 | 50 | rpcnd 13080 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐𝑄) ∈ ℂ) | 
| 52 | 1, 32 | mgpbas 20143 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) | 
| 53 | cnfldmul 21373 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 54 | 1, 53 | mgpplusg 20142 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) | 
| 55 | 52, 54 | gsumws2 18856 | . . . 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 1372 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“(𝐴↑𝑐𝑃)(𝐵↑𝑐𝑄)”〉) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) | 
| 57 | 43, 56 | eqtrd 2776 | . 2 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg (〈“𝐴𝐵”〉 ∘f ↑𝑐〈“𝑃𝑄”〉)) = ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄))) | 
| 58 | ofs2 15011 | . . . . 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 7448 | . . 3 ⊢ (𝜑 → (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉)) = (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉)) | 
| 61 | 9, 18 | rpmulcld 13094 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ+) | 
| 62 | 61 | rpcnd 13080 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ) | 
| 63 | 10, 19 | rpmulcld 13094 | . . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ+) | 
| 64 | 63 | rpcnd 13080 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ) | 
| 65 | 32, 33 | gsumws2 18856 | . . . 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 1372 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“(𝐴 · 𝑃)(𝐵 · 𝑄)”〉) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) | 
| 67 | 60, 66 | eqtrd 2776 | . 2 ⊢ (𝜑 → (ℂfld Σg (〈“𝐴𝐵”〉 ∘f · 〈“𝑃𝑄”〉)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄))) | 
| 68 | 38, 57, 67 | 3brtr3d 5173 | 1 ⊢ (𝜑 → ((𝐴↑𝑐𝑃) · (𝐵↑𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 class class class wbr 5142 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 Fincfn 8986 ℂcc 11154 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 ≤ cle 11297 ℕcn 12267 2c2 12322 ℝ+crp 13035 ..^cfzo 13695 ♯chash 14370 Word cword 14553 〈“cs2 14881 Σg cgsu 17486 Mndcmnd 18748 mulGrpcmgp 20138 Ringcrg 20231 ℂfldccnfld 21365 ↑𝑐ccxp 26598 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-word 14554 df-concat 14610 df-s1 14635 df-s2 14888 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 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 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-mulg 19087 df-subg 19142 df-ghm 19232 df-gim 19278 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-subrng 20547 df-subrg 20571 df-drng 20732 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-refld 21624 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599 df-cxp 26600 | 
| This theorem is referenced by: young2d 49379 | 
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