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 25826. (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 2736 | . . 3 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
2 | fzofi 13512 | . . . 4 ⊢ (0..^2) ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (0..^2) ∈ Fin) |
4 | 2nn 11868 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | lbfzo0 13247 | . . . . . 6 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
6 | 4, 5 | mpbir 234 | . . . . 5 ⊢ 0 ∈ (0..^2) |
7 | 6 | ne0ii 4238 | . . . 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 14401 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word ℝ+) |
12 | wrdf 14039 | . . . . 5 ⊢ (〈“𝐴𝐵”〉 ∈ Word ℝ+ → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) | |
13 | s2len 14419 | . . . . . . . 8 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
14 | 13 | eqcomi 2745 | . . . . . . 7 ⊢ 2 = (♯‘〈“𝐴𝐵”〉) |
15 | 14 | oveq2i 7202 | . . . . . 6 ⊢ (0..^2) = (0..^(♯‘〈“𝐴𝐵”〉)) |
16 | 15 | feq2i 6515 | . . . . 5 ⊢ (〈“𝐴𝐵”〉:(0..^2)⟶ℝ+ ↔ 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶ℝ+) |
17 | 12, 16 | sylibr 237 | . . . 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 25826 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉)↑𝑐(1 / (♯‘(0..^2)))) ≤ ((ℂfld Σg 〈“𝐴𝐵”〉) / (♯‘(0..^2)))) |
20 | cnring 20339 | . . . . 5 ⊢ ℂfld ∈ Ring | |
21 | 1 | ringmgp 19522 | . . . . 5 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
22 | 20, 21 | mp1i 13 | . . . 4 ⊢ (𝜑 → (mulGrp‘ℂfld) ∈ Mnd) |
23 | 9 | rpcnd 12595 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
24 | 10 | rpcnd 12595 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
25 | cnfldbas 20321 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
26 | 1, 25 | mgpbas 19464 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
27 | cnfldmul 20323 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
28 | 1, 27 | mgpplusg 19462 | . . . . 5 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
29 | 26, 28 | gsumws2 18223 | . . . 4 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉) = (𝐴 · 𝐵)) |
30 | 22, 23, 24, 29 | syl3anc 1373 | . . 3 ⊢ (𝜑 → ((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉) = (𝐴 · 𝐵)) |
31 | 2nn0 12072 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
32 | hashfzo0 13962 | . . . . 5 ⊢ (2 ∈ ℕ0 → (♯‘(0..^2)) = 2) | |
33 | 31, 32 | mp1i 13 | . . . 4 ⊢ (𝜑 → (♯‘(0..^2)) = 2) |
34 | 33 | oveq2d 7207 | . . 3 ⊢ (𝜑 → (1 / (♯‘(0..^2))) = (1 / 2)) |
35 | 30, 34 | oveq12d 7209 | . 2 ⊢ (𝜑 → (((mulGrp‘ℂfld) Σg 〈“𝐴𝐵”〉)↑𝑐(1 / (♯‘(0..^2)))) = ((𝐴 · 𝐵)↑𝑐(1 / 2))) |
36 | ringmnd 19526 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
37 | 20, 36 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ Mnd) |
38 | cnfldadd 20322 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
39 | 25, 38 | gsumws2 18223 | . . . 4 ⊢ ((ℂfld ∈ Mnd ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℂfld Σg 〈“𝐴𝐵”〉) = (𝐴 + 𝐵)) |
40 | 37, 23, 24, 39 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (ℂfld Σg 〈“𝐴𝐵”〉) = (𝐴 + 𝐵)) |
41 | 40, 33 | oveq12d 7209 | . 2 ⊢ (𝜑 → ((ℂfld Σg 〈“𝐴𝐵”〉) / (♯‘(0..^2))) = ((𝐴 + 𝐵) / 2)) |
42 | 19, 35, 41 | 3brtr3d 5070 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 class class class wbr 5039 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 ℂcc 10692 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 ≤ cle 10833 / cdiv 11454 ℕcn 11795 2c2 11850 ℕ0cn0 12055 ℝ+crp 12551 ..^cfzo 13203 ♯chash 13861 Word cword 14034 〈“cs2 14371 Σg cgsu 16899 Mndcmnd 18127 mulGrpcmgp 19458 Ringcrg 19516 ℂfldccnfld 20317 ↑𝑐ccxp 25398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-s2 14378 df-shft 14595 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ef 15592 df-sin 15594 df-cos 15595 df-pi 15597 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-mulg 18443 df-subg 18494 df-ghm 18574 df-gim 18617 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-drng 19723 df-subrg 19752 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-refld 20521 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 df-perf 21988 df-cn 22078 df-cnp 22079 df-haus 22166 df-cmp 22238 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-xms 23172 df-ms 23173 df-tms 23174 df-cncf 23729 df-limc 24717 df-dv 24718 df-log 25399 df-cxp 25400 |
This theorem is referenced by: (None) |
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