amgmwlem - Mathbox for Kunhao Zheng

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Theorem amgmwlem 48003

Description: Weighted version of amgmlem 26837. (Contributed by Kunhao Zheng, 19-Jun-2021.)

**Hypotheses**
Ref	Expression
amgmwlem.0	⊢ 𝑀 = (mulGrp‘ℂ_fld)
amgmwlem.1	⊢ (𝜑 → 𝐴 ∈ Fin)
amgmwlem.2	⊢ (𝜑 → 𝐴 ≠ ∅)
amgmwlem.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ⁺)
amgmwlem.4	⊢ (𝜑 → 𝑊:𝐴⟶ℝ⁺)
amgmwlem.5	⊢ (𝜑 → (ℂ_fld Σ_g 𝑊) = 1)

**Assertion**
Ref	Expression
amgmwlem	⊢ (𝜑 → (𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)) ≤ (ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)))

Proof of Theorem amgmwlem Dummy variables 𝑎 𝑏 𝑠 𝑢 𝑣 𝑘 𝑦 𝑤 𝑥 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		amgmwlem.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		amgmwlem.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ⁺)
3	2	ffvelcdmda 7076	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ℝ⁺)
4		amgmwlem.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊:𝐴⟶ℝ⁺)
5	4	ffvelcdmda 7076	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑊‘𝑘) ∈ ℝ⁺)
6	5	rpred 13012	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑊‘𝑘) ∈ ℝ)
7	3, 6	rpcxpcld 26582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘)) ∈ ℝ⁺)
8	7	relogcld 26472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) ∈ ℝ)
9	8	recnd 11238	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) ∈ ℂ)
10	1, 9	gsumfsum 21295	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))) = Σ𝑘 ∈ 𝐴 (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))
11	9	negnegd 11558	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → --(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) = (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))
12	11	sumeq2dv 15645	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 --(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) = Σ𝑘 ∈ 𝐴 (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))
13	8	renegcld 11637	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) ∈ ℝ)
14	13	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) ∈ ℂ)
15	1, 14	fsumneg 15729	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 --(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) = -Σ𝑘 ∈ 𝐴 -(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))
16	3, 6	logcxpd 26583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) = ((𝑊‘𝑘) · (log‘(𝐹‘𝑘))))
17	16	negeqd 11450	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) = -((𝑊‘𝑘) · (log‘(𝐹‘𝑘))))
18	17	sumeq2dv 15645	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 -(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) = Σ𝑘 ∈ 𝐴 -((𝑊‘𝑘) · (log‘(𝐹‘𝑘))))
19	18	negeqd 11450	. . . . . . . 8 ⊢ (𝜑 → -Σ𝑘 ∈ 𝐴 -(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) = -Σ𝑘 ∈ 𝐴 -((𝑊‘𝑘) · (log‘(𝐹‘𝑘))))
20	5	rpcnd 13014	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑊‘𝑘) ∈ ℂ)
21	3	relogcld 26472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (log‘(𝐹‘𝑘)) ∈ ℝ)
22	21	recnd 11238	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (log‘(𝐹‘𝑘)) ∈ ℂ)
23	20, 22	mulneg2d 11664	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))) = -((𝑊‘𝑘) · (log‘(𝐹‘𝑘))))
24	23	eqcomd 2730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -((𝑊‘𝑘) · (log‘(𝐹‘𝑘))) = ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))
25	24	sumeq2dv 15645	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 -((𝑊‘𝑘) · (log‘(𝐹‘𝑘))) = Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))
26	25	negeqd 11450	. . . . . . . 8 ⊢ (𝜑 → -Σ𝑘 ∈ 𝐴 -((𝑊‘𝑘) · (log‘(𝐹‘𝑘))) = -Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))
27	15, 19, 26	3eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 --(log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))) = -Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))
28	10, 12, 27	3eqtr2rd 2771	. . . . . 6 ⊢ (𝜑 → -Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))))
29		negex 11454	. . . . . . . . . . 11 ⊢ -(log‘(𝐹‘𝑘)) ∈ V
30	29	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘(𝐹‘𝑘)) ∈ V)
31	4	feqmptd 6950	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 = (𝑘 ∈ 𝐴 ↦ (𝑊‘𝑘)))
32		eqidd 2725	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))) = (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))
33	1, 5, 30, 31, 32	offval2 7683	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) = (𝑘 ∈ 𝐴 ↦ ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘)))))
34	33	oveq2d 7417	. . . . . . . 8 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))))
35	22	negcld 11554	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘(𝐹‘𝑘)) ∈ ℂ)
36	20, 35	mulcld 11230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))) ∈ ℂ)
37	1, 36	gsumfsum 21295	. . . . . . . 8 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))) = Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))
38	34, 37	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))) = Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))
39	38	negeqd 11450	. . . . . 6 ⊢ (𝜑 → -(ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))) = -Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · -(log‘(𝐹‘𝑘))))
40		relogf1o 26416	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
41		f1of 6823	. . . . . . . . . 10 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
42	40, 41	ax-mp 5	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺):ℝ⁺⟶ℝ
43		rpre 12978	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
44	43	anim2i 616	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺) → (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ))
45	44	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ))
46		rpcxpcl 26525	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → (𝑥↑_𝑐𝑦) ∈ ℝ⁺)
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝑥↑_𝑐𝑦) ∈ ℝ⁺)
48		inidm 4210	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐴) = 𝐴
49	47, 2, 4, 1, 1, 48	off 7681	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘_f ↑_𝑐𝑊):𝐴⟶ℝ⁺)
50		fcompt 7123	. . . . . . . . 9 ⊢ (((log ↾ ℝ⁺):ℝ⁺⟶ℝ ∧ (𝐹 ∘_f ↑_𝑐𝑊):𝐴⟶ℝ⁺) → ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊)) = (𝑘 ∈ 𝐴 ↦ ((log ↾ ℝ⁺)‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘))))
51	42, 49, 50	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊)) = (𝑘 ∈ 𝐴 ↦ ((log ↾ ℝ⁺)‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘))))
52	49	ffvelcdmda 7076	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘) ∈ ℝ⁺)
53		fvres 6900	. . . . . . . . . . 11 ⊢ (((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘) ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘)) = (log‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘)))
54	52, 53	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((log ↾ ℝ⁺)‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘)) = (log‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘)))
55	2	ffnd 6708	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn 𝐴)
56	4	ffnd 6708	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 Fn 𝐴)
57		eqidd 2725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) = (𝐹‘𝑘))
58		eqidd 2725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑊‘𝑘) = (𝑊‘𝑘))
59	55, 56, 1, 1, 48, 57, 58	ofval 7674	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘) = ((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘)))
60	59	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (log‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘)) = (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))
61	54, 60	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((log ↾ ℝ⁺)‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘)) = (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))
62	61	mpteq2dva 5238	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ ((log ↾ ℝ⁺)‘((𝐹 ∘_f ↑_𝑐𝑊)‘𝑘))) = (𝑘 ∈ 𝐴 ↦ (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘)))))
63	51, 62	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊)) = (𝑘 ∈ 𝐴 ↦ (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘)))))
64	63	oveq2d 7417	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊))) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (log‘((𝐹‘𝑘)↑_𝑐(𝑊‘𝑘))))))
65	28, 39, 64	3eqtr4d 2774	. . . . 5 ⊢ (𝜑 → -(ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))) = (ℂ_fld Σ_g ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊))))
66		amgmwlem.0	. . . . . . . . . . . . 13 ⊢ 𝑀 = (mulGrp‘ℂ_fld)
67	66	oveq1i 7411	. . . . . . . . . . . 12 ⊢ (𝑀 ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
68	67	rpmsubg 21292	. . . . . . . . . . 11 ⊢ ℝ⁺ ∈ (SubGrp‘(𝑀 ↾_s (ℂ ∖ {0})))
69		subgsubm 19064	. . . . . . . . . . 11 ⊢ (ℝ⁺ ∈ (SubGrp‘(𝑀 ↾_s (ℂ ∖ {0}))) → ℝ⁺ ∈ (SubMnd‘(𝑀 ↾_s (ℂ ∖ {0}))))
70	68, 69	ax-mp 5	. . . . . . . . . 10 ⊢ ℝ⁺ ∈ (SubMnd‘(𝑀 ↾_s (ℂ ∖ {0})))
71		cnring 21250	. . . . . . . . . . 11 ⊢ ℂ_fld ∈ Ring
72		cnfldbas 21231	. . . . . . . . . . . . 13 ⊢ ℂ = (Base‘ℂ_fld)
73		cnfld0 21252	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘ℂ_fld)
74		cndrng 21257	. . . . . . . . . . . . 13 ⊢ ℂ_fld ∈ DivRing
75	72, 73, 74	drngui 20582	. . . . . . . . . . . 12 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
76	75, 66	unitsubm 20277	. . . . . . . . . . 11 ⊢ (ℂ_fld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘𝑀))
77		eqid 2724	. . . . . . . . . . . 12 ⊢ (𝑀 ↾_s (ℂ ∖ {0})) = (𝑀 ↾_s (ℂ ∖ {0}))
78	77	subsubm 18730	. . . . . . . . . . 11 ⊢ ((ℂ ∖ {0}) ∈ (SubMnd‘𝑀) → (ℝ⁺ ∈ (SubMnd‘(𝑀 ↾_s (ℂ ∖ {0}))) ↔ (ℝ⁺ ∈ (SubMnd‘𝑀) ∧ ℝ⁺ ⊆ (ℂ ∖ {0}))))
79	71, 76, 78	mp2b 10	. . . . . . . . . 10 ⊢ (ℝ⁺ ∈ (SubMnd‘(𝑀 ↾_s (ℂ ∖ {0}))) ↔ (ℝ⁺ ∈ (SubMnd‘𝑀) ∧ ℝ⁺ ⊆ (ℂ ∖ {0})))
80	70, 79	mpbi 229	. . . . . . . . 9 ⊢ (ℝ⁺ ∈ (SubMnd‘𝑀) ∧ ℝ⁺ ⊆ (ℂ ∖ {0}))
81	80	simpli 483	. . . . . . . 8 ⊢ ℝ⁺ ∈ (SubMnd‘𝑀)
82		eqid 2724	. . . . . . . . 9 ⊢ (𝑀 ↾_s ℝ⁺) = (𝑀 ↾_s ℝ⁺)
83	82	submbas 18728	. . . . . . . 8 ⊢ (ℝ⁺ ∈ (SubMnd‘𝑀) → ℝ⁺ = (Base‘(𝑀 ↾_s ℝ⁺)))
84	81, 83	ax-mp 5	. . . . . . 7 ⊢ ℝ⁺ = (Base‘(𝑀 ↾_s ℝ⁺))
85		cnfld1 21253	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
86	66, 85	ringidval 20077	. . . . . . . 8 ⊢ 1 = (0_g‘𝑀)
87		eqid 2724	. . . . . . . . . 10 ⊢ (0_g‘𝑀) = (0_g‘𝑀)
88	82, 87	subm0 18729	. . . . . . . . 9 ⊢ (ℝ⁺ ∈ (SubMnd‘𝑀) → (0_g‘𝑀) = (0_g‘(𝑀 ↾_s ℝ⁺)))
89	81, 88	ax-mp 5	. . . . . . . 8 ⊢ (0_g‘𝑀) = (0_g‘(𝑀 ↾_s ℝ⁺))
90	86, 89	eqtri 2752	. . . . . . 7 ⊢ 1 = (0_g‘(𝑀 ↾_s ℝ⁺))
91		cncrng 21249	. . . . . . . . 9 ⊢ ℂ_fld ∈ CRing
92	66	crngmgp 20135	. . . . . . . . 9 ⊢ (ℂ_fld ∈ CRing → 𝑀 ∈ CMnd)
93	91, 92	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ CMnd)
94	82	submmnd 18727	. . . . . . . . 9 ⊢ (ℝ⁺ ∈ (SubMnd‘𝑀) → (𝑀 ↾_s ℝ⁺) ∈ Mnd)
95	81, 94	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ↾_s ℝ⁺) ∈ Mnd)
96	82	subcmn 19746	. . . . . . . 8 ⊢ ((𝑀 ∈ CMnd ∧ (𝑀 ↾_s ℝ⁺) ∈ Mnd) → (𝑀 ↾_s ℝ⁺) ∈ CMnd)
97	93, 95, 96	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑀 ↾_s ℝ⁺) ∈ CMnd)
98		resubdrg 21468	. . . . . . . . . 10 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
99	98	simpli 483	. . . . . . . . 9 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
100		df-refld 21465	. . . . . . . . . 10 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
101	100	subrgring 20465	. . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ_fld ∈ Ring)
102	99, 101	ax-mp 5	. . . . . . . 8 ⊢ ℝ_fld ∈ Ring
103		ringmnd 20137	. . . . . . . 8 ⊢ (ℝ_fld ∈ Ring → ℝ_fld ∈ Mnd)
104	102, 103	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ℝ_fld ∈ Mnd)
105	66	oveq1i 7411	. . . . . . . . . 10 ⊢ (𝑀 ↾_s ℝ⁺) = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
106	105	reloggim 26448	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpIso ℝ_fld)
107		gimghm 19178	. . . . . . . . 9 ⊢ ((log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpIso ℝ_fld) → (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpHom ℝ_fld))
108	106, 107	ax-mp 5	. . . . . . . 8 ⊢ (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpHom ℝ_fld)
109		ghmmhm 19140	. . . . . . . 8 ⊢ ((log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpHom ℝ_fld) → (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) MndHom ℝ_fld))
110	108, 109	mp1i 13	. . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) MndHom ℝ_fld))
111		1red 11211	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
112	49, 1, 111	fdmfifsupp 9368	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘_f ↑_𝑐𝑊) finSupp 1)
113	84, 90, 97, 104, 1, 110, 49, 112	gsummhm 19847	. . . . . 6 ⊢ (𝜑 → (ℝ_fld Σ_g ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊))) = ((log ↾ ℝ⁺)‘((𝑀 ↾_s ℝ⁺) Σ_g (𝐹 ∘_f ↑_𝑐𝑊))))
114		subrgsubg 20468	. . . . . . . . . 10 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
115	99, 114	ax-mp 5	. . . . . . . . 9 ⊢ ℝ ∈ (SubGrp‘ℂ_fld)
116		subgsubm 19064	. . . . . . . . 9 ⊢ (ℝ ∈ (SubGrp‘ℂ_fld) → ℝ ∈ (SubMnd‘ℂ_fld))
117	115, 116	ax-mp 5	. . . . . . . 8 ⊢ ℝ ∈ (SubMnd‘ℂ_fld)
118	117	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ (SubMnd‘ℂ_fld))
119	40, 41	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
120		fco 6731	. . . . . . . 8 ⊢ (((log ↾ ℝ⁺):ℝ⁺⟶ℝ ∧ (𝐹 ∘_f ↑_𝑐𝑊):𝐴⟶ℝ⁺) → ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊)):𝐴⟶ℝ)
121	119, 49, 120	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊)):𝐴⟶ℝ)
122	1, 118, 121, 100	gsumsubm 18749	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊))) = (ℝ_fld Σ_g ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊))))
123	81	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ∈ (SubMnd‘𝑀))
124	1, 123, 49, 82	gsumsubm 18749	. . . . . . 7 ⊢ (𝜑 → (𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)) = ((𝑀 ↾_s ℝ⁺) Σ_g (𝐹 ∘_f ↑_𝑐𝑊)))
125	124	fveq2d 6885	. . . . . 6 ⊢ (𝜑 → ((log ↾ ℝ⁺)‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))) = ((log ↾ ℝ⁺)‘((𝑀 ↾_s ℝ⁺) Σ_g (𝐹 ∘_f ↑_𝑐𝑊))))
126	113, 122, 125	3eqtr4d 2774	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((log ↾ ℝ⁺) ∘ (𝐹 ∘_f ↑_𝑐𝑊))) = ((log ↾ ℝ⁺)‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))))
127	86, 93, 1, 123, 49, 112	gsumsubmcl 19828	. . . . . 6 ⊢ (𝜑 → (𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)) ∈ ℝ⁺)
128		fvres 6900	. . . . . 6 ⊢ ((𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)) ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))) = (log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))))
129	127, 128	syl 17	. . . . 5 ⊢ (𝜑 → ((log ↾ ℝ⁺)‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))) = (log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))))
130	65, 126, 129	3eqtrd 2768	. . . 4 ⊢ (𝜑 → -(ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))) = (log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))))
131		simprl 768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → 𝑥 ∈ ℝ⁺)
132	131	rpcnd 13014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → 𝑥 ∈ ℂ)
133		simprr 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
134	133	rpcnd 13014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℂ)
135	132, 134	mulcomd 11231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
136	1, 4, 2, 135	caofcom 7698	. . . . . . . 8 ⊢ (𝜑 → (𝑊 ∘_f · 𝐹) = (𝐹 ∘_f · 𝑊))
137	136	oveq2d 7417	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) = (ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)))
138	2	feqmptd 6950	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
139	1, 5, 3, 31, 138	offval2 7683	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ∘_f · 𝐹) = (𝑘 ∈ 𝐴 ↦ ((𝑊‘𝑘) · (𝐹‘𝑘))))
140	139	oveq2d 7417	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ ((𝑊‘𝑘) · (𝐹‘𝑘)))))
141	5, 3	rpmulcld 13028	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑊‘𝑘) · (𝐹‘𝑘)) ∈ ℝ⁺)
142	141	rpcnd 13014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑊‘𝑘) · (𝐹‘𝑘)) ∈ ℂ)
143	1, 142	gsumfsum 21295	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ ((𝑊‘𝑘) · (𝐹‘𝑘)))) = Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · (𝐹‘𝑘)))
144	140, 143	eqtrd 2764	. . . . . . . 8 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) = Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · (𝐹‘𝑘)))
145		amgmwlem.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ ∅)
146	1, 145, 141	fsumrpcl 15679	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((𝑊‘𝑘) · (𝐹‘𝑘)) ∈ ℝ⁺)
147	144, 146	eqeltrd 2825	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) ∈ ℝ⁺)
148	137, 147	eqeltrrd 2826	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)) ∈ ℝ⁺)
149	148	relogcld 26472	. . . . 5 ⊢ (𝜑 → (log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))) ∈ ℝ)
150		ringcmn 20170	. . . . . . 7 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
151	71, 150	mp1i 13	. . . . . 6 ⊢ (𝜑 → ℂ_fld ∈ CMnd)
152		remulcl 11190	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
153	152	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
154		rpssre 12977	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
155		fss 6724	. . . . . . . 8 ⊢ ((𝑊:𝐴⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → 𝑊:𝐴⟶ℝ)
156	4, 154, 155	sylancl 585	. . . . . . 7 ⊢ (𝜑 → 𝑊:𝐴⟶ℝ)
157	21	renegcld 11637	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘(𝐹‘𝑘)) ∈ ℝ)
158	157	fmpttd 7106	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))):𝐴⟶ℝ)
159	153, 156, 158, 1, 1, 48	off 7681	. . . . . 6 ⊢ (𝜑 → (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))):𝐴⟶ℝ)
160		0red 11213	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
161	159, 1, 160	fdmfifsupp 9368	. . . . . 6 ⊢ (𝜑 → (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) finSupp 0)
162	73, 151, 1, 118, 159, 161	gsumsubmcl 19828	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))) ∈ ℝ)
163	154	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
164		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
165	164	relogcld 26472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (log‘𝑤) ∈ ℝ)
166	165	renegcld 11637	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → -(log‘𝑤) ∈ ℝ)
167	166	fmpttd 7106	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)):ℝ⁺⟶ℝ)
168		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑎 ∈ ℝ⁺)
169		ioorp 13398	. . . . . . . . . . . 12 ⊢ (0(,)+∞) = ℝ⁺
170	168, 169	eleqtrrdi 2836	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑎 ∈ (0(,)+∞))
171		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑏 ∈ ℝ⁺)
172	171, 169	eleqtrrdi 2836	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑏 ∈ (0(,)+∞))
173		iccssioo2 13393	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (0(,)+∞) ∧ 𝑏 ∈ (0(,)+∞)) → (𝑎[,]𝑏) ⊆ (0(,)+∞))
174	170, 172, 173	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → (𝑎[,]𝑏) ⊆ (0(,)+∞))
175	174, 169	sseqtrdi 4024	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → (𝑎[,]𝑏) ⊆ ℝ⁺)
176	175	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) → (𝑎[,]𝑏) ⊆ ℝ⁺)
177		ioossico 13411	. . . . . . . . . 10 ⊢ (0(,)+∞) ⊆ (0[,)+∞)
178	169, 177	eqsstrri 4009	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ (0[,)+∞)
179		fss 6724	. . . . . . . . 9 ⊢ ((𝑊:𝐴⟶ℝ⁺ ∧ ℝ⁺ ⊆ (0[,)+∞)) → 𝑊:𝐴⟶(0[,)+∞))
180	4, 178, 179	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → 𝑊:𝐴⟶(0[,)+∞))
181		0lt1 11732	. . . . . . . . 9 ⊢ 0 < 1
182		amgmwlem.5	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g 𝑊) = 1)
183	181, 182	breqtrrid 5176	. . . . . . . 8 ⊢ (𝜑 → 0 < (ℂ_fld Σ_g 𝑊))
184		logccv 26512	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))) < (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
185	184	3adant1 1127	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))) < (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
186		elioore 13350	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (0(,)1) → 𝑡 ∈ ℝ)
187	186	3ad2ant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑡 ∈ ℝ)
188		simp21 1203	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑥 ∈ ℝ⁺)
189	188	relogcld 26472	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘𝑥) ∈ ℝ)
190	187, 189	remulcld 11240	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · (log‘𝑥)) ∈ ℝ)
191		1red 11211	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (0(,)1) → 1 ∈ ℝ)
192	191, 186	resubcld 11638	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (0(,)1) → (1 − 𝑡) ∈ ℝ)
193	192	3ad2ant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (1 − 𝑡) ∈ ℝ)
194		simp22 1204	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑦 ∈ ℝ⁺)
195	194	relogcld 26472	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘𝑦) ∈ ℝ)
196	193, 195	remulcld 11240	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · (log‘𝑦)) ∈ ℝ)
197	190, 196	readdcld 11239	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))) ∈ ℝ)
198		eliooord 13379	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (0(,)1) → (0 < 𝑡 ∧ 𝑡 < 1))
199	198	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (0(,)1) → 0 < 𝑡)
200	186, 199	elrpd 13009	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (0(,)1) → 𝑡 ∈ ℝ⁺)
201	200	3ad2ant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑡 ∈ ℝ⁺)
202	201, 188	rpmulcld 13028	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · 𝑥) ∈ ℝ⁺)
203		0red 11213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (0(,)1) → 0 ∈ ℝ)
204	198	simprd 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ (0(,)1) → 𝑡 < 1)
205		1m0e1 12329	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 − 0) = 1
206	204, 205	breqtrrdi 5180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (0(,)1) → 𝑡 < (1 − 0))
207	186, 191, 203, 206	ltsub13d 11816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (0(,)1) → 0 < (1 − 𝑡))
208	192, 207	elrpd 13009	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (0(,)1) → (1 − 𝑡) ∈ ℝ⁺)
209	208	3ad2ant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (1 − 𝑡) ∈ ℝ⁺)
210	209, 194	rpmulcld 13028	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · 𝑦) ∈ ℝ⁺)
211		rpaddcl 12992	. . . . . . . . . . . . . 14 ⊢ (((𝑡 · 𝑥) ∈ ℝ⁺ ∧ ((1 − 𝑡) · 𝑦) ∈ ℝ⁺) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ ℝ⁺)
212	202, 210, 211	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ ℝ⁺)
213	212	relogcld 26472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ ℝ)
214	197, 213	ltnegd 11788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))) < (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ↔ -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) < -((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦)))))
215	185, 214	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) < -((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))))
216		eqidd 2725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) = (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)))
217		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) → (log‘𝑤) = (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
218	217	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) ∧ 𝑤 = ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (log‘𝑤) = (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
219	218	negeqd 11450	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) ∧ 𝑤 = ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → -(log‘𝑤) = -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
220		negex 11454	. . . . . . . . . . . 12 ⊢ -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ V
221	220	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ V)
222	216, 219, 212, 221	fvmptd 6995	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
223		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (log‘𝑤) = (log‘𝑥))
224	223	negeqd 11450	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑥 → -(log‘𝑤) = -(log‘𝑥))
225		eqid 2724	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) = (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))
226		negex 11454	. . . . . . . . . . . . . . . 16 ⊢ -(log‘𝑤) ∈ V
227	224, 225, 226	fvmpt3i 6993	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥) = -(log‘𝑥))
228	188, 227	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥) = -(log‘𝑥))
229	228	oveq2d 7417	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥)) = (𝑡 · -(log‘𝑥)))
230	187	recnd 11238	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑡 ∈ ℂ)
231	189	recnd 11238	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘𝑥) ∈ ℂ)
232	230, 231	mulneg2d 11664	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · -(log‘𝑥)) = -(𝑡 · (log‘𝑥)))
233	229, 232	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥)) = -(𝑡 · (log‘𝑥)))
234		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑦 → (log‘𝑤) = (log‘𝑦))
235	234	negeqd 11450	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → -(log‘𝑤) = -(log‘𝑦))
236	235, 225, 226	fvmpt3i 6993	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ⁺ → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦) = -(log‘𝑦))
237	194, 236	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦) = -(log‘𝑦))
238	237	oveq2d 7417	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦)) = ((1 − 𝑡) · -(log‘𝑦)))
239	209	rpcnd 13014	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (1 − 𝑡) ∈ ℂ)
240	195	recnd 11238	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘𝑦) ∈ ℂ)
241	239, 240	mulneg2d 11664	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · -(log‘𝑦)) = -((1 − 𝑡) · (log‘𝑦)))
242	238, 241	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦)) = -((1 − 𝑡) · (log‘𝑦)))
243	233, 242	oveq12d 7419	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥)) + ((1 − 𝑡) · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦))) = (-(𝑡 · (log‘𝑥)) + -((1 − 𝑡) · (log‘𝑦))))
244	190	recnd 11238	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · (log‘𝑥)) ∈ ℂ)
245	196	recnd 11238	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · (log‘𝑦)) ∈ ℂ)
246	244, 245	negdid 11580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → -((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))) = (-(𝑡 · (log‘𝑥)) + -((1 − 𝑡) · (log‘𝑦))))
247	243, 246	eqtr4d 2767	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥)) + ((1 − 𝑡) · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦))) = -((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))))
248	215, 222, 247	3brtr4d 5170	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) < ((𝑡 · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥)) + ((1 − 𝑡) · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦))))
249	163, 167, 176, 248	scvxcvx 26833	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺ ∧ 𝑠 ∈ (0[,]1))) → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((𝑠 · 𝑢) + ((1 − 𝑠) · 𝑣))) ≤ ((𝑠 · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑢)) + ((1 − 𝑠) · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑣))))
250	163, 167, 176, 1, 180, 2, 183, 249	jensen 26836	. . . . . . 7 ⊢ (𝜑 → (((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / (ℂ_fld Σ_g 𝑊)) ∈ ℝ⁺ ∧ ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / (ℂ_fld Σ_g 𝑊))) ≤ ((ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / (ℂ_fld Σ_g 𝑊))))
251	250	simprd 495	. . . . . 6 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / (ℂ_fld Σ_g 𝑊))) ≤ ((ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / (ℂ_fld Σ_g 𝑊)))
252	182	oveq2d 7417	. . . . . . . 8 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / (ℂ_fld Σ_g 𝑊)) = ((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / 1))
253	252	fveq2d 6885	. . . . . . 7 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / (ℂ_fld Σ_g 𝑊))) = ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / 1)))
254	147	rpcnd 13014	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) ∈ ℂ)
255	254	div1d 11978	. . . . . . . 8 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / 1) = (ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)))
256	255	fveq2d 6885	. . . . . . 7 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / 1)) = ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))))
257		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑤 = (ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) → (log‘𝑤) = (log‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))))
258	257	negeqd 11450	. . . . . . . . . 10 ⊢ (𝑤 = (ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) → -(log‘𝑤) = -(log‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))))
259	258, 225, 226	fvmpt3i 6993	. . . . . . . . 9 ⊢ ((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) ∈ ℝ⁺ → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))) = -(log‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))))
260	147, 259	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))) = -(log‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))))
261	137	fveq2d 6885	. . . . . . . . 9 ⊢ (𝜑 → (log‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))) = (log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))))
262	261	negeqd 11450	. . . . . . . 8 ⊢ (𝜑 → -(log‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))) = -(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))))
263	260, 262	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘(ℂ_fld Σ_g (𝑊 ∘_f · 𝐹))) = -(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))))
264	253, 256, 263	3eqtrd 2768	. . . . . 6 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g (𝑊 ∘_f · 𝐹)) / (ℂ_fld Σ_g 𝑊))) = -(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))))
265	182	oveq2d 7417	. . . . . . 7 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / (ℂ_fld Σ_g 𝑊)) = ((ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / 1))
266		ringmnd 20137	. . . . . . . . . . 11 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
267	71, 266	ax-mp 5	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Mnd
268	72	submid 18724	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Mnd → ℂ ∈ (SubMnd‘ℂ_fld))
269	267, 268	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ℂ ∈ (SubMnd‘ℂ_fld))
270		mulcl 11189	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
271	270	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
272		rpcn 12980	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
273	272	ssriv 3978	. . . . . . . . . . . 12 ⊢ ℝ⁺ ⊆ ℂ
274	273	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℝ⁺ ⊆ ℂ)
275	4, 274	fssd 6725	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊:𝐴⟶ℂ)
276	165	recnd 11238	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (log‘𝑤) ∈ ℂ)
277	276	negcld 11554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → -(log‘𝑤) ∈ ℂ)
278	277	fmpttd 7106	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)):ℝ⁺⟶ℂ)
279		fco 6731	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)):ℝ⁺⟶ℂ ∧ 𝐹:𝐴⟶ℝ⁺) → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹):𝐴⟶ℂ)
280	278, 2, 279	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹):𝐴⟶ℂ)
281	271, 275, 280, 1, 1, 48	off 7681	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹)):𝐴⟶ℂ)
282	281, 1, 160	fdmfifsupp 9368	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹)) finSupp 0)
283	73, 151, 1, 269, 281, 282	gsumsubmcl 19828	. . . . . . . 8 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) ∈ ℂ)
284	283	div1d 11978	. . . . . . 7 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / 1) = (ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))))
285		eqidd 2725	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) = (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)))
286		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹‘𝑘) → (log‘𝑤) = (log‘(𝐹‘𝑘)))
287	286	negeqd 11450	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹‘𝑘) → -(log‘𝑤) = -(log‘(𝐹‘𝑘)))
288	3, 138, 285, 287	fmptco 7119	. . . . . . . . 9 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))
289	288	oveq2d 7417	. . . . . . . 8 ⊢ (𝜑 → (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹)) = (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))))
290	289	oveq2d 7417	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) = (ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))))
291	265, 284, 290	3eqtrd 2768	. . . . . 6 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑊 ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / (ℂ_fld Σ_g 𝑊)) = (ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))))
292	251, 264, 291	3brtr3d 5169	. . . . 5 ⊢ (𝜑 → -(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))) ≤ (ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))))
293	149, 162, 292	lenegcon1d 11792	. . . 4 ⊢ (𝜑 → -(ℂ_fld Σ_g (𝑊 ∘_f · (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))) ≤ (log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))))
294	130, 293	eqbrtrrd 5162	. . 3 ⊢ (𝜑 → (log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))) ≤ (log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))))
295	127	relogcld 26472	. . . 4 ⊢ (𝜑 → (log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))) ∈ ℝ)
296		efle 16057	. . . 4 ⊢ (((log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))) ∈ ℝ ∧ (log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))) ∈ ℝ) → ((log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))) ≤ (log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))) ↔ (exp‘(log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)))) ≤ (exp‘(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))))))
297	295, 149, 296	syl2anc 583	. . 3 ⊢ (𝜑 → ((log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊))) ≤ (log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))) ↔ (exp‘(log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)))) ≤ (exp‘(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊))))))
298	294, 297	mpbid 231	. 2 ⊢ (𝜑 → (exp‘(log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)))) ≤ (exp‘(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)))))
299	127	reeflogd 26473	. . 3 ⊢ (𝜑 → (exp‘(log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)))) = (𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)))
300	299	eqcomd 2730	. 2 ⊢ (𝜑 → (𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)) = (exp‘(log‘(𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)))))
301	148	reeflogd 26473	. . 3 ⊢ (𝜑 → (exp‘(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)))) = (ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)))
302	301	eqcomd 2730	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)) = (exp‘(log‘(ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)))))
303	298, 300, 302	3brtr4d 5170	1 ⊢ (𝜑 → (𝑀 Σ_g (𝐹 ∘_f ↑_𝑐𝑊)) ≤ (ℂ_fld Σ_g (𝐹 ∘_f · 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ∖ cdif 3937 ⊆ wss 3940 ∅c0 4314 {csn 4620 class class class wbr 5138 ↦ cmpt 5221 ↾ cres 5668 ∘ ccom 5670 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7401 ∘_f cof 7661 Fincfn 8934 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 +∞cpnf 11241 < clt 11244 ≤ cle 11245 − cmin 11440 -cneg 11441 / cdiv 11867 ℝ⁺crp 12970 (,)cioo 13320 [,)cico 13322 [,]cicc 13323 Σcsu 15628 expce 16001 Basecbs 17142 ↾_s cress 17171 0_gc0g 17383 Σ_g cgsu 17384 Mndcmnd 18656 MndHom cmhm 18700 SubMndcsubmnd 18701 SubGrpcsubg 19036 GrpHom cghm 19127 GrpIso cgim 19171 CMndccmn 19689 mulGrpcmgp 20028 Ringcrg 20127 CRingccrg 20128 SubRingcsubrg 20458 DivRingcdr 20576 ℂ_fldccnfld 21227 ℝ_fldcrefld 21464 logclog 26404 ↑_𝑐ccxp 26405

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-mhm 18702 df-submnd 18703 df-grp 18855 df-minusg 18856 df-mulg 18985 df-subg 19039 df-ghm 19128 df-gim 19173 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-subrng 20435 df-subrg 20460 df-drng 20578 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-refld 21465 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-cmp 23212 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 df-log 26406 df-cxp 26407

This theorem is referenced by: amgmlemALT 48004 amgmw2d 48005

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