amgmw2d - Mathbox for Kunhao Zheng

	Mathbox for Kunhao Zheng		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > amgmw2d		Structured version Visualization version GIF version

Theorem amgmw2d 50933

Description: Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 45157). (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 2761	. . 3 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
2		fzofi 14097	. . . 4 ⊢ (0..^2) ∈ Fin
3	2	a1i 11	. . 3 ⊢ (𝜑 → (0..^2) ∈ Fin)
4		2nn 12397	. . . . 5 ⊢ 2 ∈ ℕ
5		lbfzo0 13814	. . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
6	4, 5	mpbir 234	. . . 4 ⊢ 0 ∈ (0..^2)
7		ne0i 4287	. . . 4 ⊢ (0 ∈ (0..^2) → (0..^2) ≠ ∅)
8	6, 7	mp1i 14	. . 3 ⊢ (𝜑 → (0..^2) ≠ ∅)
9		amgmw2d.0	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
10		amgmw2d.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
11	9, 10	s2cld 15002	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺)
12		wrdf 14643	. . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺ → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
13	11, 12	syl 18	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
14		s2len 15020	. . . . . 6 ⊢ (♯‘⟨“𝐴𝐵”⟩) = 2
15	14	oveq2i 7423	. . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵”⟩)) = (0..^2)
16	15	feq2i 6693	. . . 4 ⊢ (⟨“𝐴𝐵”⟩:(0..^(♯‘⟨“𝐴𝐵”⟩))⟶ℝ⁺ ↔ ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ⁺)
17	13, 16	sylib 221	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ⁺)
18		amgmw2d.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
19		amgmw2d.3	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ⁺)
20	18, 19	s2cld 15002	. . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺)
21		wrdf 14643	. . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺ → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
22	20, 21	syl 18	. . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
23		s2len 15020	. . . . . 6 ⊢ (♯‘⟨“𝑃𝑄”⟩) = 2
24	23	oveq2i 7423	. . . . 5 ⊢ (0..^(♯‘⟨“𝑃𝑄”⟩)) = (0..^2)
25	24	feq2i 6693	. . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(♯‘⟨“𝑃𝑄”⟩))⟶ℝ⁺ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
26	22, 25	sylib 221	. . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
27		cnring 21680	. . . . . 6 ⊢ ℂ_fld ∈ Ring
28		ringmnd 20450	. . . . . 6 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
29	27, 28	mp1i 14	. . . . 5 ⊢ (𝜑 → ℂ_fld ∈ Mnd)
30	18	rpcnd 13147	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
31	19	rpcnd 13147	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ)
32		cnfldbas 21662	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
33		cnfldadd 21664	. . . . . 6 ⊢ + = (+_g‘ℂ_fld)
34	32, 33	gsumws2 19018	. . . . 5 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
35	29, 30, 31, 34	syl3anc 1398	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
36		amgmw2d.4	. . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1)
37	35, 36	eqtrd 2796	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = 1)
38	1, 3, 8, 17, 26, 37	amgmwlem 50931	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) ≤ (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)))
39	9, 10	jca 521	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺))
40	18, 19	jca 521	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺))
41		ofs2 15104	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺)) → (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩) = ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩)
42	39, 40, 41	syl2anc 596	. . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩) = ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩)
43	42	oveq2d 7428	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩))
44	1	ringmgp 20445	. . . . 5 ⊢ (ℂ_fld ∈ Ring → (mulGrp‘ℂ_fld) ∈ Mnd)
45	27, 44	mp1i 14	. . . 4 ⊢ (𝜑 → (mulGrp‘ℂ_fld) ∈ Mnd)
46	18	rpred 13145	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ)
47	9, 46	rpcxpcld 27043	. . . . 5 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℝ⁺)
48	47	rpcnd 13147	. . . 4 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℂ)
49	19	rpred 13145	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ)
50	10, 49	rpcxpcld 27043	. . . . 5 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℝ⁺)
51	50	rpcnd 13147	. . . 4 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℂ)
52	1, 32	mgpbas 20345	. . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
53		cnfldmul 21666	. . . . . 6 ⊢ · = (._r‘ℂ_fld)
54	1, 53	mgpplusg 20344	. . . . 5 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
55	52, 54	gsumws2 19018	. . . 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 1398	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
57	43, 56	eqtrd 2796	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_f ↑_𝑐⟨“𝑃𝑄”⟩)) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
58		ofs2 15104	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺)) → (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)
59	39, 40, 58	syl2anc 596	. . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)
60	59	oveq2d 7428	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_f · ⟨“𝑃𝑄”⟩)) = (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩))
61	9, 18	rpmulcld 13161	. . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ⁺)
62	61	rpcnd 13147	. . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ)
63	10, 19	rpmulcld 13161	. . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ⁺)
64	63	rpcnd 13147	. . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ)
65	32, 33	gsumws2 19018	. . . 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 1398	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
67	60, 66	eqtrd 2796	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∅c0 4279 class class class wbr 5103 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ∘_f cof 7680 Fincfn 8957 ℂcc 11179 0cc0 11181 1c1 11182 + caddc 11184 · cmul 11186 ≤ cle 11325 ℕcn 12316 2c2 12378 ℝ⁺crp 13101 ..^cfzo 13768 ♯chash 14454 Word cword 14638 ⟨“cs2 14972 Σ_g cgsu 17591 Mndcmnd 18903 mulGrpcmgp 20340 Ringcrg 20439 ℂ_fldccnfld 21658 ↑_𝑐ccxp 26865

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260 ax-mulf 11261

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-word 14639 df-concat 14696 df-s1 14723 df-s2 14979 df-shft 15200 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-sin 16215 df-cos 16216 df-pi 16218 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-mhm 18958 df-submnd 18959 df-grp 19127 df-minusg 19128 df-mulg 19258 df-subg 19313 df-ghm 19408 df-gim 19453 df-cntz 19511 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-cring 20442 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-dvr 20611 df-subrng 20778 df-subrg 20802 df-drng 20962 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-refld 21891 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-cmp 23685 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-limc 26166 df-dv 26167 df-log 26866 df-cxp 26867

This theorem is used by: young2d 50934

W3C validator