amgmw2d - Mathbox for Kunhao Zheng

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Theorem amgmw2d 42359

Description: Weighted arithmetic-geometric mean inequality for 𝑛 = 2 (compare amgm2d 37523). (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 2610	. . 3 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
2		fzofi 12635	. . . 4 ⊢ (0..^2) ∈ Fin
3	2	a1i 11	. . 3 ⊢ (𝜑 → (0..^2) ∈ Fin)
4		2nn 11062	. . . . 5 ⊢ 2 ∈ ℕ
5		lbfzo0 12375	. . . . 5 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
6	4, 5	mpbir 220	. . . 4 ⊢ 0 ∈ (0..^2)
7		ne0i 3880	. . . 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 13466	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺)
12		wrdf 13165	. . . . 5 ⊢ (⟨“𝐴𝐵”⟩ ∈ Word ℝ⁺ → ⟨“𝐴𝐵”⟩:(0..^(#‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^(#‘⟨“𝐴𝐵”⟩))⟶ℝ⁺)
14		s2len 13484	. . . . . 6 ⊢ (#‘⟨“𝐴𝐵”⟩) = 2
15	14	oveq2i 6560	. . . . 5 ⊢ (0..^(#‘⟨“𝐴𝐵”⟩)) = (0..^2)
16	15	feq2i 5950	. . . 4 ⊢ (⟨“𝐴𝐵”⟩:(0..^(#‘⟨“𝐴𝐵”⟩))⟶ℝ⁺ ↔ ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ⁺)
17	13, 16	sylib 207	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵”⟩:(0..^2)⟶ℝ⁺)
18		amgmw2d.1	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
19		amgmw2d.3	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ⁺)
20	18, 19	s2cld 13466	. . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺)
21		wrdf 13165	. . . . 5 ⊢ (⟨“𝑃𝑄”⟩ ∈ Word ℝ⁺ → ⟨“𝑃𝑄”⟩:(0..^(#‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
22	20, 21	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^(#‘⟨“𝑃𝑄”⟩))⟶ℝ⁺)
23		s2len 13484	. . . . . 6 ⊢ (#‘⟨“𝑃𝑄”⟩) = 2
24	23	oveq2i 6560	. . . . 5 ⊢ (0..^(#‘⟨“𝑃𝑄”⟩)) = (0..^2)
25	24	feq2i 5950	. . . 4 ⊢ (⟨“𝑃𝑄”⟩:(0..^(#‘⟨“𝑃𝑄”⟩))⟶ℝ⁺ ↔ ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
26	22, 25	sylib 207	. . 3 ⊢ (𝜑 → ⟨“𝑃𝑄”⟩:(0..^2)⟶ℝ⁺)
27		cnring 19587	. . . . . 6 ⊢ ℂ_fld ∈ Ring
28		ringmnd 18379	. . . . . 6 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
29	27, 28	mp1i 13	. . . . 5 ⊢ (𝜑 → ℂ_fld ∈ Mnd)
30	18	rpcnd 11750	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
31	19	rpcnd 11750	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℂ)
32		cnfldbas 19571	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
33		cnfldadd 19572	. . . . . 6 ⊢ + = (+_g‘ℂ_fld)
34	32, 33	gsumws2 17202	. . . . 5 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ) → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
35	29, 30, 31, 34	syl3anc 1318	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = (𝑃 + 𝑄))
36		amgmw2d.4	. . . 4 ⊢ (𝜑 → (𝑃 + 𝑄) = 1)
37	35, 36	eqtrd 2644	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“𝑃𝑄”⟩) = 1)
38	1, 3, 8, 17, 26, 37	amgmwlem 42357	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_𝑓 ↑_𝑐⟨“𝑃𝑄”⟩)) ≤ (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_𝑓 · ⟨“𝑃𝑄”⟩)))
39	9, 10	jca 553	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺))
40	18, 19	jca 553	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺))
41		ofs2 13558	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺)) → (⟨“𝐴𝐵”⟩ ∘_𝑓 ↑_𝑐⟨“𝑃𝑄”⟩) = ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩)
42	39, 40, 41	syl2anc 691	. . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘_𝑓 ↑_𝑐⟨“𝑃𝑄”⟩) = ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩)
43	42	oveq2d 6565	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_𝑓 ↑_𝑐⟨“𝑃𝑄”⟩)) = ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩))
44		eqid 2610	. . . . . 6 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
45	44	ringmgp 18376	. . . . 5 ⊢ (ℂ_fld ∈ Ring → (mulGrp‘ℂ_fld) ∈ Mnd)
46	27, 45	mp1i 13	. . . 4 ⊢ (𝜑 → (mulGrp‘ℂ_fld) ∈ Mnd)
47	18	rpred 11748	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ)
48	9, 47	rpcxpcld 24276	. . . . 5 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℝ⁺)
49	48	rpcnd 11750	. . . 4 ⊢ (𝜑 → (𝐴↑_𝑐𝑃) ∈ ℂ)
50	19	rpred 11748	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ)
51	10, 50	rpcxpcld 24276	. . . . 5 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℝ⁺)
52	51	rpcnd 11750	. . . 4 ⊢ (𝜑 → (𝐵↑_𝑐𝑄) ∈ ℂ)
53		eqid 2610	. . . . . 6 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
54	53, 32	mgpbas 18318	. . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
55		eqid 2610	. . . . . 6 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
56		cnfldmul 19573	. . . . . 6 ⊢ · = (._r‘ℂ_fld)
57	55, 56	mgpplusg 18316	. . . . 5 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
58	54, 57	gsumws2 17202	. . . 4 ⊢ (((mulGrp‘ℂ_fld) ∈ Mnd ∧ (𝐴↑_𝑐𝑃) ∈ ℂ ∧ (𝐵↑_𝑐𝑄) ∈ ℂ) → ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
59	46, 49, 52, 58	syl3anc 1318	. . 3 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g ⟨“(𝐴↑_𝑐𝑃)(𝐵↑_𝑐𝑄)”⟩) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
60	43, 59	eqtrd 2644	. 2 ⊢ (𝜑 → ((mulGrp‘ℂ_fld) Σ_g (⟨“𝐴𝐵”⟩ ∘_𝑓 ↑_𝑐⟨“𝑃𝑄”⟩)) = ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)))
61	9, 10	jca 553	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺))
62	18, 19	jca 553	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺))
63		ofs2 13558	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) ∧ (𝑃 ∈ ℝ⁺ ∧ 𝑄 ∈ ℝ⁺)) → (⟨“𝐴𝐵”⟩ ∘_𝑓 · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)
64	61, 62, 63	syl2anc 691	. . . 4 ⊢ (𝜑 → (⟨“𝐴𝐵”⟩ ∘_𝑓 · ⟨“𝑃𝑄”⟩) = ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩)
65	64	oveq2d 6565	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_𝑓 · ⟨“𝑃𝑄”⟩)) = (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩))
66		ringmnd 18379	. . . . 5 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
67	27, 66	mp1i 13	. . . 4 ⊢ (𝜑 → ℂ_fld ∈ Mnd)
68	9, 18	rpmulcld 11764	. . . . 5 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℝ⁺)
69	68	rpcnd 11750	. . . 4 ⊢ (𝜑 → (𝐴 · 𝑃) ∈ ℂ)
70	10, 19	rpmulcld 11764	. . . . 5 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℝ⁺)
71	70	rpcnd 11750	. . . 4 ⊢ (𝜑 → (𝐵 · 𝑄) ∈ ℂ)
72	32, 33	gsumws2 17202	. . . 4 ⊢ ((ℂ_fld ∈ Mnd ∧ (𝐴 · 𝑃) ∈ ℂ ∧ (𝐵 · 𝑄) ∈ ℂ) → (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
73	67, 69, 71, 72	syl3anc 1318	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g ⟨“(𝐴 · 𝑃)(𝐵 · 𝑄)”⟩) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
74	65, 73	eqtrd 2644	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (⟨“𝐴𝐵”⟩ ∘_𝑓 · ⟨“𝑃𝑄”⟩)) = ((𝐴 · 𝑃) + (𝐵 · 𝑄)))
75	38, 60, 74	3brtr3d 4614	1 ⊢ (𝜑 → ((𝐴↑_𝑐𝑃) · (𝐵↑_𝑐𝑄)) ≤ ((𝐴 · 𝑃) + (𝐵 · 𝑄)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 Fincfn 7841 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 ℕcn 10897 2c2 10947 ℝ⁺crp 11708 ..^cfzo 12334 #chash 12979 Word cword 13146 ⟨“cs2 13437 Σ_g cgsu 15924 Mndcmnd 17117 mulGrpcmgp 18312 Ringcrg 18370 ℂ_fldccnfld 19567 ↑_𝑐ccxp 24106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-subg 17414 df-ghm 17481 df-gim 17524 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-subrg 18601 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-refld 19770 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-cxp 24108

This theorem is referenced by: young2d 42360

W3C validator