amgmw2d - Mathbox for Kunhao Zheng

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Theorem amgmw2d 47241

Description: Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 42461). (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 13879	. . . 4 ⊢ (0..^2) ∈ Fin
3	2	a1i 11	. . 3 ⊢ (𝜑 → (0..^2) ∈ Fin)
4		2nn 12226	. . . . 5 ⊢ 2 ∈ ℕ
5		lbfzo0 13612	. . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
6	4, 5	mpbir 230	. . . 4 ⊢ 0 ∈ (0..^2)
7		ne0i 4294	. . . 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 14760	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺)
12		wrdf 14407	. . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
14		s2len 14778	. . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2
15	14	oveq2i 7368	. . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2)
16	15	feq2i 6660	. . . 4 ⊢ (⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺ ↔ ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ⁺)
17	13, 16	sylib 217	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ⁺)
18		amgmw2d.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
19		amgmw2d.3	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ⁺)
20	18, 19	s2cld 14760	. . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺)
21		wrdf 14407	. . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
22	20, 21	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
23		s2len 14778	. . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2
24	23	oveq2i 7368	. . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2)
25	24	feq2i 6660	. . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
26	22, 25	sylib 217	. . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
27		cnring 20819	. . . . . 6 ⊢ ℂ_fld ∈ Ring
28		ringmnd 19974	. . . . . 6 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
29	27, 28	mp1i 13	. . . . 5 ⊢ (𝜑 → ℂ_fld ∈ Mnd)
30	18	rpcnd 12959	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
31	19	rpcnd 12959	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ)
32		cnfldbas 20800	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
33		cnfldadd 20801	. . . . . 6 ⊢ + = (+_g‘ℂ_fld)
34	32, 33	gsumws2 18652	. . . . 5 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
35	29, 30, 31, 34	syl3anc 1371	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
36		amgmw2d.4	. . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1)
37	35, 36	eqtrd 2776	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = 1)
38	1, 3, 8, 17, 26, 37	amgmwlem 47239	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) ≤ (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)))
39	9, 10	jca 512	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺))
40	18, 19	jca 512	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺))
41		ofs2 14856	. . . . 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 7373	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩))
44	1	ringmgp 19970	. . . . 5 ⊢ (ℂ_fld ∈ Ring → (mulGrp‘ℂ_fld) ∈ Mnd)
45	27, 44	mp1i 13	. . . 4 ⊢ (𝜑 → (mulGrp‘ℂ_fld) ∈ Mnd)
46	18	rpred 12957	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ)
47	9, 46	rpcxpcld 26087	. . . . 5 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℝ⁺)
48	47	rpcnd 12959	. . . 4 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℂ)
49	19	rpred 12957	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ)
50	10, 49	rpcxpcld 26087	. . . . 5 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℝ⁺)
51	50	rpcnd 12959	. . . 4 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℂ)
52	1, 32	mgpbas 19902	. . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
53		cnfldmul 20802	. . . . . 6 ⊢ · = (._r‘ℂ_fld)
54	1, 53	mgpplusg 19900	. . . . 5 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
55	52, 54	gsumws2 18652	. . . 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 1371	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
57	43, 56	eqtrd 2776	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
58		ofs2 14856	. . . . 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 7373	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)) = (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩))
61	9, 18	rpmulcld 12973	. . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ⁺)
62	61	rpcnd 12959	. . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ)
63	10, 19	rpmulcld 12973	. . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ⁺)
64	63	rpcnd 12959	. . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ)
65	32, 33	gsumws2 18652	. . . 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 1371	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
67	60, 66	eqtrd 2776	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
68	38, 57, 67	3brtr3d 5136	1 ⊢ (𝜑 → ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∅c0 4282 class class class wbr 5105 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∘_f cof 7615 Fincfn 8883 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 ≤ cle 11190 ℕcn 12153 2c2 12208 ℝ⁺crp 12915 ..^cfzo 13567 ♯chash 14230 Word cword 14402 ⟨“cs2 14730 Σ_g cgsu 17322 Mndcmnd 18556 mulGrpcmgp 19896 Ringcrg 19964 ℂ_fldccnfld 20796 ↑_𝑐ccxp 25911

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-s2 14737 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-mulg 18873 df-subg 18925 df-ghm 19006 df-gim 19049 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-subrg 20220 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-refld 21009 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-cxp 25913

This theorem is referenced by: young2d 47242

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