amgmw2d - Mathbox for Kunhao Zheng

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Theorem amgmw2d 50279

Description: Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 44625). (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 2736	. . 3 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
2		fzofi 13936	. . . 4 ⊢ (0..^2) ∈ Fin
3	2	a1i 11	. . 3 ⊢ (𝜑 → (0..^2) ∈ Fin)
4		2nn 12254	. . . . 5 ⊢ 2 ∈ ℕ
5		lbfzo0 13654	. . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
6	4, 5	mpbir 231	. . . 4 ⊢ 0 ∈ (0..^2)
7		ne0i 4281	. . . 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 14833	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺)
12		wrdf 14480	. . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
14		s2len 14851	. . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2
15	14	oveq2i 7378	. . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2)
16	15	feq2i 6660	. . . 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 14833	. . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺)
21		wrdf 14480	. . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
22	20, 21	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
23		s2len 14851	. . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2
24	23	oveq2i 7378	. . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2)
25	24	feq2i 6660	. . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
26	22, 25	sylib 218	. . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
27		cnring 21374	. . . . . 6 ⊢ ℂ_fld ∈ Ring
28		ringmnd 20224	. . . . . 6 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
29	27, 28	mp1i 13	. . . . 5 ⊢ (𝜑 → ℂ_fld ∈ Mnd)
30	18	rpcnd 12988	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
31	19	rpcnd 12988	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ)
32		cnfldbas 21356	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
33		cnfldadd 21358	. . . . . 6 ⊢ + = (+_g‘ℂ_fld)
34	32, 33	gsumws2 18810	. . . . 5 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
35	29, 30, 31, 34	syl3anc 1374	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
36		amgmw2d.4	. . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1)
37	35, 36	eqtrd 2771	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = 1)
38	1, 3, 8, 17, 26, 37	amgmwlem 50277	. 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 14933	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺)) → (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩) = ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩)
42	39, 40, 41	syl2anc 585	. . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩) = ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩)
43	42	oveq2d 7383	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩))
44	1	ringmgp 20220	. . . . 5 ⊢ (ℂ_fld ∈ Ring → (mulGrp‘ℂ_fld) ∈ Mnd)
45	27, 44	mp1i 13	. . . 4 ⊢ (𝜑 → (mulGrp‘ℂ_fld) ∈ Mnd)
46	18	rpred 12986	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ)
47	9, 46	rpcxpcld 26697	. . . . 5 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℝ⁺)
48	47	rpcnd 12988	. . . 4 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℂ)
49	19	rpred 12986	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ)
50	10, 49	rpcxpcld 26697	. . . . 5 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℝ⁺)
51	50	rpcnd 12988	. . . 4 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℂ)
52	1, 32	mgpbas 20126	. . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
53		cnfldmul 21360	. . . . . 6 ⊢ · = (._r‘ℂ_fld)
54	1, 53	mgpplusg 20125	. . . . 5 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
55	52, 54	gsumws2 18810	. . . 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 1374	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
57	43, 56	eqtrd 2771	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
58		ofs2 14933	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺)) → (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)
59	39, 40, 58	syl2anc 585	. . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)
60	59	oveq2d 7383	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)) = (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩))
61	9, 18	rpmulcld 13002	. . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ⁺)
62	61	rpcnd 12988	. . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ)
63	10, 19	rpmulcld 13002	. . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ⁺)
64	63	rpcnd 12988	. . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ)
65	32, 33	gsumws2 18810	. . . 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 1374	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
67	60, 66	eqtrd 2771	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
68	38, 57, 67	3brtr3d 5116	1 ⊢ (𝜑 → ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∅c0 4273 class class class wbr 5085 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∘_f cof 7629 Fincfn 8893 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ≤ cle 11180 ℕcn 12174 2c2 12236 ℝ⁺crp 12942 ..^cfzo 13608 ♯chash 14292 Word cword 14475 ⟨“cs2 14803 Σ_g cgsu 17403 Mndcmnd 18702 mulGrpcmgp 20121 Ringcrg 20214 ℂ_fldccnfld 21352 ↑_𝑐ccxp 26519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-word 14476 df-concat 14533 df-s1 14559 df-s2 14810 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-mulg 19044 df-subg 19099 df-ghm 19188 df-gim 19234 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-subrng 20523 df-subrg 20547 df-drng 20708 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-refld 21585 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-cxp 26521

This theorem is referenced by: young2d 50280

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