amgmw2d - Mathbox for Kunhao Zheng

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Theorem amgmw2d 49835

Description: Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 44230). (Contributed by Kunhao Zheng, 20-Jun-2021.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
amgmw2d	⊢ (𝜑 → ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄)))

**Proof of Theorem amgmw2d**
Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
2		fzofi 13878	. . . 4 ⊢ (0..^2) ∈ Fin
3	2	a1i 11	. . 3 ⊢ (𝜑 → (0..^2) ∈ Fin)
4		2nn 12195	. . . . 5 ⊢ 2 ∈ ℕ
5		lbfzo0 13596	. . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
6	4, 5	mpbir 231	. . . 4 ⊢ 0 ∈ (0..^2)
7		ne0i 4291	. . . 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 14775	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺)
12		wrdf 14422	. . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
14		s2len 14793	. . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2
15	14	oveq2i 7357	. . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2)
16	15	feq2i 6643	. . . 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 14775	. . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺)
21		wrdf 14422	. . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
22	20, 21	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
23		s2len 14793	. . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2
24	23	oveq2i 7357	. . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2)
25	24	feq2i 6643	. . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
26	22, 25	sylib 218	. . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
27		cnring 21325	. . . . . 6 ⊢ ℂ_fld ∈ Ring
28		ringmnd 20159	. . . . . 6 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
29	27, 28	mp1i 13	. . . . 5 ⊢ (𝜑 → ℂ_fld ∈ Mnd)
30	18	rpcnd 12933	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
31	19	rpcnd 12933	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ)
32		cnfldbas 21293	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
33		cnfldadd 21295	. . . . . 6 ⊢ + = (+_g‘ℂ_fld)
34	32, 33	gsumws2 18747	. . . . 5 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
35	29, 30, 31, 34	syl3anc 1373	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
36		amgmw2d.4	. . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1)
37	35, 36	eqtrd 2766	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = 1)
38	1, 3, 8, 17, 26, 37	amgmwlem 49833	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) ≤ (ℂ_fld Σ_g (⟨“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 14875	. . . . 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 7362	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩))
44	1	ringmgp 20155	. . . . 5 ⊢ (ℂ_fld ∈ Ring → (mulGrp‘ℂ_fld) ∈ Mnd)
45	27, 44	mp1i 13	. . . 4 ⊢ (𝜑 → (mulGrp‘ℂ_fld) ∈ Mnd)
46	18	rpred 12931	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ)
47	9, 46	rpcxpcld 26667	. . . . 5 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℝ⁺)
48	47	rpcnd 12933	. . . 4 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℂ)
49	19	rpred 12931	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ)
50	10, 49	rpcxpcld 26667	. . . . 5 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℝ⁺)
51	50	rpcnd 12933	. . . 4 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℂ)
52	1, 32	mgpbas 20061	. . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
53		cnfldmul 21297	. . . . . 6 ⊢ · = (._r‘ℂ_fld)
54	1, 53	mgpplusg 20060	. . . . 5 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
55	52, 54	gsumws2 18747	. . . 4 ⊢ (((mulGrp‘ℂ_fld) ∈ Mnd ∧ (𝐴↑_𝑐𝑃) ∈ ℂ ∧ (𝐵↑_𝑐𝑄) ∈ ℂ) → ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
56	45, 48, 51, 55	syl3anc 1373	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
57	43, 56	eqtrd 2766	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
58		ofs2 14875	. . . . 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 7362	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)) = (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩))
61	9, 18	rpmulcld 12947	. . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ⁺)
62	61	rpcnd 12933	. . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ)
63	10, 19	rpmulcld 12947	. . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ⁺)
64	63	rpcnd 12933	. . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ)
65	32, 33	gsumws2 18747	. . . 4 ⊢ ((ℂ_fld ∈ Mnd ∧ (𝐴 · 𝑃) ∈ ℂ ∧ (𝐵 · 𝑄) ∈ ℂ) → (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
66	29, 62, 64, 65	syl3anc 1373	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
67	60, 66	eqtrd 2766	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
68	38, 57, 67	3brtr3d 5122	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 2111 ≠ wne 2928 ∅c0 4283 class class class wbr 5091 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∘_f cof 7608 Fincfn 8869 ℂcc 11001 0cc0 11003 1c1 11004 + caddc 11006 · cmul 11008 ≤ cle 11144 ℕcn 12122 2c2 12177 ℝ⁺crp 12887 ..^cfzo 13551 ♯chash 14234 Word cword 14417 ⟨“cs2 14745 Σ_g cgsu 17341 Mndcmnd 18639 mulGrpcmgp 20056 Ringcrg 20149 ℂ_fldccnfld 21289 ↑_𝑐ccxp 26489

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082 ax-mulf 11083

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-ioc 13247 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-fl 13693 df-mod 13771 df-seq 13906 df-exp 13966 df-fac 14178 df-bc 14207 df-hash 14235 df-word 14418 df-concat 14475 df-s1 14501 df-s2 14752 df-shft 14971 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-limsup 15375 df-clim 15392 df-rlim 15393 df-sum 15591 df-ef 15971 df-sin 15973 df-cos 15974 df-pi 15976 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-pt 17345 df-prds 17348 df-xrs 17403 df-qtop 17408 df-imas 17409 df-xps 17411 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-mhm 18688 df-submnd 18689 df-grp 18846 df-minusg 18847 df-mulg 18978 df-subg 19033 df-ghm 19123 df-gim 19169 df-cntz 19227 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-subrng 20459 df-subrg 20483 df-drng 20644 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-fbas 21286 df-fg 21287 df-cnfld 21290 df-refld 21540 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-cld 22932 df-ntr 22933 df-cls 22934 df-nei 23011 df-lp 23049 df-perf 23050 df-cn 23140 df-cnp 23141 df-haus 23228 df-cmp 23300 df-tx 23475 df-hmeo 23668 df-fil 23759 df-fm 23851 df-flim 23852 df-flf 23853 df-xms 24233 df-ms 24234 df-tms 24235 df-cncf 24796 df-limc 25792 df-dv 25793 df-log 26490 df-cxp 26491

This theorem is referenced by: young2d 49836

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