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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > amgm2d | Structured version Visualization version GIF version |
Description: Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 26935. (Contributed by Stanislas Polu, 8-Sep-2020.) |
Ref | Expression |
---|---|
amgm2d.0 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
amgm2d.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
amgm2d | ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13972 | . . . 4 ⊢ (0..^2) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
4 | 2nn 12316 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | lbfzo0 13705 | . . . . . 6 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
6 | 4, 5 | mpbir 230 | . . . . 5 ⊢ 0 ∈ (0..^2) |
7 | 6 | ne0ii 4338 | . . . 4 ⊢ (0..^2) ≠ ∅ |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ≠ ∅) |
9 | amgm2d.0 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | amgm2d.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
11 | 9, 10 | s2cld 14855 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word ℝ+) |
12 | wrdf 14502 | . . . . 5 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) | |
13 | s2len 14873 | . . . . . . . 8 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
14 | 13 | eqcomi 2737 | . . . . . . 7 ⊢ 2 = (♯‘〈“𝐴𝐵”〉) |
15 | 14 | oveq2i 7431 | . . . . . 6 ⊢ (0..^2) = (0..^(♯‘〈“𝐴𝐵”〉)) |
16 | 15 | feq2i 6714 | . . . . 5 ⊢ (〈“𝐴𝐵”〉:(0..^2)⟶ℝ+ ↔ 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) |
17 | 12, 16 | sylibr 233 | . . . 4 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
18 | 11, 17 | syl 17 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉:(0..^2)⟶ℝ+) |
19 | 1, 3, 8, 18 | amgmlem 26935 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉)↑𝑐(1 / (♯‘(0..^2)))) ≤ ((ℂfld Σg 〈“𝐴𝐵”〉) / (♯‘(0..^2)))) |
20 | cnring 21318 | . . . . 5 ⊢ ℂfld ∈ Ring | |
21 | 1 | ringmgp 20179 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
22 | 20, 21 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
23 | 9 | rpcnd 13051 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
24 | 10 | rpcnd 13051 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
25 | cnfldbas 21283 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
26 | 1, 25 | mgpbas 20080 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
27 | cnfldmul 21287 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
28 | 1, 27 | mgpplusg 20078 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
29 | 26, 28 | gsumws2 18794 | . . . 4 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉) = (𝐴 · 𝐵)) |
30 | 22, 23, 24, 29 | syl3anc 1369 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉) = (𝐴 · 𝐵)) |
31 | 2nn0 12520 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
32 | hashfzo0 14422 | . . . . 5 ⊢ (2 ∈ ℕ0 → (♯‘(0..^2)) = 2) | |
33 | 31, 32 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^2)) = 2) |
34 | 33 | oveq2d 7436 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^2))) = (1 / 2)) |
35 | 30, 34 | oveq12d 7438 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉)↑𝑐(1 / (♯‘(0..^2)))) = ((𝐴 · 𝐵)↑𝑐(1 / 2))) |
36 | ringmnd 20183 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
37 | 20, 36 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
38 | cnfldadd 21285 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
39 | 25, 38 | gsumws2 18794 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℂfld Σg 〈“𝐴𝐵”〉) = (𝐴 + 𝐵)) |
40 | 37, 23, 24, 39 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝐴𝐵”〉) = (𝐴 + 𝐵)) |
41 | 40, 33 | oveq12d 7438 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵”〉) / (♯‘(0..^2))) = ((𝐴 + 𝐵) / 2)) |
42 | 19, 35, 41 | 3brtr3d 5179 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∅c0 4323 class class class wbr 5148 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 Fincfn 8964 ℂcc 11137 0cc0 11139 1c1 11140 + caddc 11142 · cmul 11144 ≤ cle 11280 / cdiv 11902 ℕcn 12243 2c2 12298 ℕ0cn0 12503 ℝ+crp 13007 ..^cfzo 13660 ♯chash 14322 Word cword 14497 〈“cs2 14825 Σg cgsu 17422 Mndcmnd 18694 mulGrpcmgp 20074 Ringcrg 20173 ℂfldccnfld 21279 ↑𝑐ccxp 26502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-word 14498 df-concat 14554 df-s1 14579 df-s2 14832 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-pi 16049 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-mulg 19024 df-subg 19078 df-ghm 19168 df-gim 19213 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-subrng 20483 df-subrg 20508 df-drng 20626 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-refld 21537 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-cmp 23304 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503 df-cxp 26504 |
This theorem is referenced by: (None) |
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