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Theorem amgmlem 26718

Description: Lemma for amgm 26719. (Contributed by Mario Carneiro, 21-Jun-2015.)

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

**Assertion**
Ref	Expression
amgmlem	⊢ (𝜑 → ((𝑀 Σ_g 𝐹)↑_𝑐(1 / (♯‘𝐴))) ≤ ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))

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

Step	Hyp	Ref	Expression
1		cnfld0 21169	. . . . . . . 8 ⊢ 0 = (0_g‘ℂ_fld)
2		cnring 21167	. . . . . . . . 9 ⊢ ℂ_fld ∈ Ring
3		ringabl 20169	. . . . . . . . 9 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Abel)
4	2, 3	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → ℂ_fld ∈ Abel)
5		amgm.2	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		resubdrg 21380	. . . . . . . . . 10 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
7	6	simpli 484	. . . . . . . . 9 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
8		subrgsubg 20467	. . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
9	7, 8	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ (SubGrp‘ℂ_fld))
10		amgm.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ⁺)
11	10	ffvelcdmda 7086	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ℝ⁺)
12	11	relogcld 26355	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (log‘(𝐹‘𝑘)) ∈ ℝ)
13	12	renegcld 11645	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘(𝐹‘𝑘)) ∈ ℝ)
14	13	fmpttd 7116	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))):𝐴⟶ℝ)
15		c0ex 11212	. . . . . . . . . 10 ⊢ 0 ∈ V
16	15	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
17	14, 5, 16	fdmfifsupp 9375	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))) finSupp 0)
18	1, 4, 5, 9, 14, 17	gsumsubgcl 19829	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) ∈ ℝ)
19	18	recnd 11246	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) ∈ ℂ)
20		amgm.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ ∅)
21		hashnncl 14330	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
22	5, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
23	20, 22	mpbird 256	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ)
24	23	nncnd 12232	. . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ)
25	23	nnne0d 12266	. . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ≠ 0)
26	19, 24, 25	divnegd 12007	. . . . 5 ⊢ (𝜑 → -((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) = (-(ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)))
27	12	recnd 11246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (log‘(𝐹‘𝑘)) ∈ ℂ)
28	5, 27	gsumfsum 21212	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (log‘(𝐹‘𝑘)))) = Σ𝑘 ∈ 𝐴 (log‘(𝐹‘𝑘)))
29	27	negnegd 11566	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → --(log‘(𝐹‘𝑘)) = (log‘(𝐹‘𝑘)))
30	29	sumeq2dv 15653	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 --(log‘(𝐹‘𝑘)) = Σ𝑘 ∈ 𝐴 (log‘(𝐹‘𝑘)))
31	13	recnd 11246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘(𝐹‘𝑘)) ∈ ℂ)
32	5, 31	fsumneg 15737	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 --(log‘(𝐹‘𝑘)) = -Σ𝑘 ∈ 𝐴 -(log‘(𝐹‘𝑘)))
33	28, 30, 32	3eqtr2rd 2779	. . . . . . . 8 ⊢ (𝜑 → -Σ𝑘 ∈ 𝐴 -(log‘(𝐹‘𝑘)) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (log‘(𝐹‘𝑘)))))
34	5, 31	gsumfsum 21212	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) = Σ𝑘 ∈ 𝐴 -(log‘(𝐹‘𝑘)))
35	34	negeqd 11458	. . . . . . . 8 ⊢ (𝜑 → -(ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) = -Σ𝑘 ∈ 𝐴 -(log‘(𝐹‘𝑘)))
36	10	feqmptd 6960	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
37		relogf1o 26299	. . . . . . . . . . . . 13 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
38		f1of 6833	. . . . . . . . . . . . 13 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
39	37, 38	mp1i 13	. . . . . . . . . . . 12 ⊢ (𝜑 → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
40	39	feqmptd 6960	. . . . . . . . . . 11 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
41		fvres 6910	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
42	41	mpteq2ia 5251	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
43	40, 42	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
44		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑘) → (log‘𝑥) = (log‘(𝐹‘𝑘)))
45	11, 36, 43, 44	fmptco 7129	. . . . . . . . 9 ⊢ (𝜑 → ((log ↾ ℝ⁺) ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (log‘(𝐹‘𝑘))))
46	45	oveq2d 7427	. . . . . . . 8 ⊢ (𝜑 → (ℂ_fld Σ_g ((log ↾ ℝ⁺) ∘ 𝐹)) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (log‘(𝐹‘𝑘)))))
47	33, 35, 46	3eqtr4d 2782	. . . . . . 7 ⊢ (𝜑 → -(ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) = (ℂ_fld Σ_g ((log ↾ ℝ⁺) ∘ 𝐹)))
48		amgm.1	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = (mulGrp‘ℂ_fld)
49	48	oveq1i 7421	. . . . . . . . . . . . . 14 ⊢ (𝑀 ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
50	49	rpmsubg 21209	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ∈ (SubGrp‘(𝑀 ↾_s (ℂ ∖ {0})))
51		subgsubm 19064	. . . . . . . . . . . . 13 ⊢ (ℝ⁺ ∈ (SubGrp‘(𝑀 ↾_s (ℂ ∖ {0}))) → ℝ⁺ ∈ (SubMnd‘(𝑀 ↾_s (ℂ ∖ {0}))))
52	50, 51	ax-mp 5	. . . . . . . . . . . 12 ⊢ ℝ⁺ ∈ (SubMnd‘(𝑀 ↾_s (ℂ ∖ {0})))
53		cnfldbas 21148	. . . . . . . . . . . . . . 15 ⊢ ℂ = (Base‘ℂ_fld)
54		cndrng 21174	. . . . . . . . . . . . . . 15 ⊢ ℂ_fld ∈ DivRing
55	53, 1, 54	drngui 20506	. . . . . . . . . . . . . 14 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
56	55, 48	unitsubm 20277	. . . . . . . . . . . . 13 ⊢ (ℂ_fld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘𝑀))
57		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝑀 ↾_s (ℂ ∖ {0})) = (𝑀 ↾_s (ℂ ∖ {0}))
58	57	subsubm 18733	. . . . . . . . . . . . 13 ⊢ ((ℂ ∖ {0}) ∈ (SubMnd‘𝑀) → (ℝ⁺ ∈ (SubMnd‘(𝑀 ↾_s (ℂ ∖ {0}))) ↔ (ℝ⁺ ∈ (SubMnd‘𝑀) ∧ ℝ⁺ ⊆ (ℂ ∖ {0}))))
59	2, 56, 58	mp2b 10	. . . . . . . . . . . 12 ⊢ (ℝ⁺ ∈ (SubMnd‘(𝑀 ↾_s (ℂ ∖ {0}))) ↔ (ℝ⁺ ∈ (SubMnd‘𝑀) ∧ ℝ⁺ ⊆ (ℂ ∖ {0})))
60	52, 59	mpbi 229	. . . . . . . . . . 11 ⊢ (ℝ⁺ ∈ (SubMnd‘𝑀) ∧ ℝ⁺ ⊆ (ℂ ∖ {0}))
61	60	simpli 484	. . . . . . . . . 10 ⊢ ℝ⁺ ∈ (SubMnd‘𝑀)
62		eqid 2732	. . . . . . . . . . 11 ⊢ (𝑀 ↾_s ℝ⁺) = (𝑀 ↾_s ℝ⁺)
63	62	submbas 18731	. . . . . . . . . 10 ⊢ (ℝ⁺ ∈ (SubMnd‘𝑀) → ℝ⁺ = (Base‘(𝑀 ↾_s ℝ⁺)))
64	61, 63	ax-mp 5	. . . . . . . . 9 ⊢ ℝ⁺ = (Base‘(𝑀 ↾_s ℝ⁺))
65		cnfld1 21170	. . . . . . . . . . . 12 ⊢ 1 = (1_r‘ℂ_fld)
66	48, 65	ringidval 20077	. . . . . . . . . . 11 ⊢ 1 = (0_g‘𝑀)
67	62, 66	subm0 18732	. . . . . . . . . 10 ⊢ (ℝ⁺ ∈ (SubMnd‘𝑀) → 1 = (0_g‘(𝑀 ↾_s ℝ⁺)))
68	61, 67	ax-mp 5	. . . . . . . . 9 ⊢ 1 = (0_g‘(𝑀 ↾_s ℝ⁺))
69		cncrng 21166	. . . . . . . . . . 11 ⊢ ℂ_fld ∈ CRing
70	48	crngmgp 20135	. . . . . . . . . . 11 ⊢ (ℂ_fld ∈ CRing → 𝑀 ∈ CMnd)
71	69, 70	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ CMnd)
72	62	submmnd 18730	. . . . . . . . . . 11 ⊢ (ℝ⁺ ∈ (SubMnd‘𝑀) → (𝑀 ↾_s ℝ⁺) ∈ Mnd)
73	61, 72	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 ↾_s ℝ⁺) ∈ Mnd)
74	62	subcmn 19746	. . . . . . . . . 10 ⊢ ((𝑀 ∈ CMnd ∧ (𝑀 ↾_s ℝ⁺) ∈ Mnd) → (𝑀 ↾_s ℝ⁺) ∈ CMnd)
75	71, 73, 74	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ↾_s ℝ⁺) ∈ CMnd)
76		df-refld 21377	. . . . . . . . . . . 12 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
77	76	subrgring 20464	. . . . . . . . . . 11 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ_fld ∈ Ring)
78	7, 77	ax-mp 5	. . . . . . . . . 10 ⊢ ℝ_fld ∈ Ring
79		ringmnd 20137	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ Ring → ℝ_fld ∈ Mnd)
80	78, 79	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ℝ_fld ∈ Mnd)
81	48	oveq1i 7421	. . . . . . . . . . . 12 ⊢ (𝑀 ↾_s ℝ⁺) = ((mulGrp‘ℂ_fld) ↾_s ℝ⁺)
82	81	reloggim 26331	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpIso ℝ_fld)
83		gimghm 19178	. . . . . . . . . . 11 ⊢ ((log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpIso ℝ_fld) → (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpHom ℝ_fld))
84	82, 83	ax-mp 5	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpHom ℝ_fld)
85		ghmmhm 19140	. . . . . . . . . 10 ⊢ ((log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) GrpHom ℝ_fld) → (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) MndHom ℝ_fld))
86	84, 85	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (log ↾ ℝ⁺) ∈ ((𝑀 ↾_s ℝ⁺) MndHom ℝ_fld))
87		1ex 11214	. . . . . . . . . . 11 ⊢ 1 ∈ V
88	87	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ V)
89	10, 5, 88	fdmfifsupp 9375	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 finSupp 1)
90	64, 68, 75, 80, 5, 86, 10, 89	gsummhm 19847	. . . . . . . 8 ⊢ (𝜑 → (ℝ_fld Σ_g ((log ↾ ℝ⁺) ∘ 𝐹)) = ((log ↾ ℝ⁺)‘((𝑀 ↾_s ℝ⁺) Σ_g 𝐹)))
91		subgsubm 19064	. . . . . . . . . 10 ⊢ (ℝ ∈ (SubGrp‘ℂ_fld) → ℝ ∈ (SubMnd‘ℂ_fld))
92	9, 91	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ (SubMnd‘ℂ_fld))
93		fco 6741	. . . . . . . . . 10 ⊢ (((log ↾ ℝ⁺):ℝ⁺⟶ℝ ∧ 𝐹:𝐴⟶ℝ⁺) → ((log ↾ ℝ⁺) ∘ 𝐹):𝐴⟶ℝ)
94	39, 10, 93	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((log ↾ ℝ⁺) ∘ 𝐹):𝐴⟶ℝ)
95	5, 92, 94, 76	gsumsubm 18752	. . . . . . . 8 ⊢ (𝜑 → (ℂ_fld Σ_g ((log ↾ ℝ⁺) ∘ 𝐹)) = (ℝ_fld Σ_g ((log ↾ ℝ⁺) ∘ 𝐹)))
96	61	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ⁺ ∈ (SubMnd‘𝑀))
97	5, 96, 10, 62	gsumsubm 18752	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 Σ_g 𝐹) = ((𝑀 ↾_s ℝ⁺) Σ_g 𝐹))
98	97	fveq2d 6895	. . . . . . . 8 ⊢ (𝜑 → ((log ↾ ℝ⁺)‘(𝑀 Σ_g 𝐹)) = ((log ↾ ℝ⁺)‘((𝑀 ↾_s ℝ⁺) Σ_g 𝐹)))
99	90, 95, 98	3eqtr4d 2782	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((log ↾ ℝ⁺) ∘ 𝐹)) = ((log ↾ ℝ⁺)‘(𝑀 Σ_g 𝐹)))
100	66, 71, 5, 96, 10, 89	gsumsubmcl 19828	. . . . . . . 8 ⊢ (𝜑 → (𝑀 Σ_g 𝐹) ∈ ℝ⁺)
101	100	fvresd 6911	. . . . . . 7 ⊢ (𝜑 → ((log ↾ ℝ⁺)‘(𝑀 Σ_g 𝐹)) = (log‘(𝑀 Σ_g 𝐹)))
102	47, 99, 101	3eqtrd 2776	. . . . . 6 ⊢ (𝜑 → -(ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) = (log‘(𝑀 Σ_g 𝐹)))
103	102	oveq1d 7426	. . . . 5 ⊢ (𝜑 → (-(ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) = ((log‘(𝑀 Σ_g 𝐹)) / (♯‘𝐴)))
104	100	relogcld 26355	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑀 Σ_g 𝐹)) ∈ ℝ)
105	104	recnd 11246	. . . . . 6 ⊢ (𝜑 → (log‘(𝑀 Σ_g 𝐹)) ∈ ℂ)
106	105, 24, 25	divrec2d 11998	. . . . 5 ⊢ (𝜑 → ((log‘(𝑀 Σ_g 𝐹)) / (♯‘𝐴)) = ((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹))))
107	26, 103, 106	3eqtrd 2776	. . . 4 ⊢ (𝜑 → -((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) = ((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹))))
108	36	oveq2d 7427	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g 𝐹) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))))
109	11	rpcnd 13022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ℂ)
110	5, 109	gsumfsum 21212	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑘))
111	108, 110	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → (ℂ_fld Σ_g 𝐹) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑘))
112	5, 20, 11	fsumrpcl 15687	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ ℝ⁺)
113	111, 112	eqeltrd 2833	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g 𝐹) ∈ ℝ⁺)
114	23	nnrpd 13018	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℝ⁺)
115	113, 114	rpdivcld 13037	. . . . . 6 ⊢ (𝜑 → ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) ∈ ℝ⁺)
116	115	relogcld 26355	. . . . 5 ⊢ (𝜑 → (log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))) ∈ ℝ)
117	18, 23	nndivred 12270	. . . . 5 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) ∈ ℝ)
118		rpssre 12985	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
119	118	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
120		relogcl 26308	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℝ⁺ → (log‘𝑤) ∈ ℝ)
121	120	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (log‘𝑤) ∈ ℝ)
122	121	renegcld 11645	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → -(log‘𝑤) ∈ ℝ)
123	122	fmpttd 7116	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)):ℝ⁺⟶ℝ)
124		ioorp 13406	. . . . . . . . . . . 12 ⊢ (0(,)+∞) = ℝ⁺
125	124	eleq2i 2825	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (0(,)+∞) ↔ 𝑎 ∈ ℝ⁺)
126	124	eleq2i 2825	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (0(,)+∞) ↔ 𝑏 ∈ ℝ⁺)
127		iccssioo2 13401	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (0(,)+∞) ∧ 𝑏 ∈ (0(,)+∞)) → (𝑎[,]𝑏) ⊆ (0(,)+∞))
128	125, 126, 127	syl2anbr 599	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → (𝑎[,]𝑏) ⊆ (0(,)+∞))
129	128, 124	sseqtrdi 4032	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → (𝑎[,]𝑏) ⊆ ℝ⁺)
130	129	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) → (𝑎[,]𝑏) ⊆ ℝ⁺)
131	23	nnrecred 12267	. . . . . . . . . 10 ⊢ (𝜑 → (1 / (♯‘𝐴)) ∈ ℝ)
132	114	rpreccld 13030	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / (♯‘𝐴)) ∈ ℝ⁺)
133	132	rpge0d 13024	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (1 / (♯‘𝐴)))
134		elrege0 13435	. . . . . . . . . 10 ⊢ ((1 / (♯‘𝐴)) ∈ (0[,)+∞) ↔ ((1 / (♯‘𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (♯‘𝐴))))
135	131, 133, 134	sylanbrc 583	. . . . . . . . 9 ⊢ (𝜑 → (1 / (♯‘𝐴)) ∈ (0[,)+∞))
136		fconst6g 6780	. . . . . . . . 9 ⊢ ((1 / (♯‘𝐴)) ∈ (0[,)+∞) → (𝐴 × {(1 / (♯‘𝐴))}):𝐴⟶(0[,)+∞))
137	135, 136	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐴 × {(1 / (♯‘𝐴))}):𝐴⟶(0[,)+∞))
138		0lt1 11740	. . . . . . . . 9 ⊢ 0 < 1
139		fconstmpt 5738	. . . . . . . . . . 11 ⊢ (𝐴 × {(1 / (♯‘𝐴))}) = (𝑘 ∈ 𝐴 ↦ (1 / (♯‘𝐴)))
140	139	oveq2i 7422	. . . . . . . . . 10 ⊢ (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))})) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (1 / (♯‘𝐴))))
141		ringmnd 20137	. . . . . . . . . . . . 13 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
142	2, 141	mp1i 13	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ_fld ∈ Mnd)
143	131	recnd 11246	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 / (♯‘𝐴)) ∈ ℂ)
144		eqid 2732	. . . . . . . . . . . . 13 ⊢ (._g‘ℂ_fld) = (._g‘ℂ_fld)
145	53, 144	gsumconst 19843	. . . . . . . . . . . 12 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝐴 ∈ Fin ∧ (1 / (♯‘𝐴)) ∈ ℂ) → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (1 / (♯‘𝐴)))) = ((♯‘𝐴)(._g‘ℂ_fld)(1 / (♯‘𝐴))))
146	142, 5, 143, 145	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (1 / (♯‘𝐴)))) = ((♯‘𝐴)(._g‘ℂ_fld)(1 / (♯‘𝐴))))
147	23	nnzd 12589	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ)
148		cnfldmulg 21177	. . . . . . . . . . . 12 ⊢ (((♯‘𝐴) ∈ ℤ ∧ (1 / (♯‘𝐴)) ∈ ℂ) → ((♯‘𝐴)(._g‘ℂ_fld)(1 / (♯‘𝐴))) = ((♯‘𝐴) · (1 / (♯‘𝐴))))
149	147, 143, 148	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝐴)(._g‘ℂ_fld)(1 / (♯‘𝐴))) = ((♯‘𝐴) · (1 / (♯‘𝐴))))
150	24, 25	recidd 11989	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝐴) · (1 / (♯‘𝐴))) = 1)
151	146, 149, 150	3eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (1 / (♯‘𝐴)))) = 1)
152	140, 151	eqtrid 2784	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))})) = 1)
153	138, 152	breqtrrid 5186	. . . . . . . 8 ⊢ (𝜑 → 0 < (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))})))
154		logccv 26395	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))) < (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
155	154	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))) < (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
156		ioossre 13389	. . . . . . . . . . . . . . 15 ⊢ (0(,)1) ⊆ ℝ
157		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑡 ∈ (0(,)1))
158	156, 157	sselid 3980	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑡 ∈ ℝ)
159		simp21 1206	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑥 ∈ ℝ⁺)
160	159	relogcld 26355	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘𝑥) ∈ ℝ)
161	158, 160	remulcld 11248	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · (log‘𝑥)) ∈ ℝ)
162		1re 11218	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
163		resubcl 11528	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (1 − 𝑡) ∈ ℝ)
164	162, 158, 163	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (1 − 𝑡) ∈ ℝ)
165		simp22 1207	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑦 ∈ ℝ⁺)
166	165	relogcld 26355	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘𝑦) ∈ ℝ)
167	164, 166	remulcld 11248	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · (log‘𝑦)) ∈ ℝ)
168	161, 167	readdcld 11247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · (log‘𝑥)) + ((1 − 𝑡) · (log‘𝑦))) ∈ ℝ)
169		simp1 1136	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝜑)
170		ioossicc 13414	. . . . . . . . . . . . . . 15 ⊢ (0(,)1) ⊆ (0[,]1)
171	170, 157	sselid 3980	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑡 ∈ (0[,]1))
172	119, 130	cvxcl 26713	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ ℝ⁺)
173	169, 159, 165, 171, 172	syl13anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ ℝ⁺)
174	173	relogcld 26355	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ ℝ)
175	168, 174	ltnegd 11796	. . . . . . . . . . 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)))))
176	155, 175	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))))
177		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) → (log‘𝑤) = (log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
178	177	negeqd 11458	. . . . . . . . . . . 12 ⊢ (𝑤 = ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) → -(log‘𝑤) = -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
179		eqid 2732	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) = (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))
180		negex 11462	. . . . . . . . . . . 12 ⊢ -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ V
181	178, 179, 180	fvmpt 6998	. . . . . . . . . . 11 ⊢ (((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ ℝ⁺ → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = -(log‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))))
182	173, 181	syl 17	. . . . . . . . . 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))))
183		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (log‘𝑤) = (log‘𝑥))
184	183	negeqd 11458	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑥 → -(log‘𝑤) = -(log‘𝑥))
185		negex 11462	. . . . . . . . . . . . . . . 16 ⊢ -(log‘𝑥) ∈ V
186	184, 179, 185	fvmpt 6998	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥) = -(log‘𝑥))
187	159, 186	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥) = -(log‘𝑥))
188	187	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥)) = (𝑡 · -(log‘𝑥)))
189	158	recnd 11246	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → 𝑡 ∈ ℂ)
190	160	recnd 11246	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘𝑥) ∈ ℂ)
191	189, 190	mulneg2d 11672	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · -(log‘𝑥)) = -(𝑡 · (log‘𝑥)))
192	188, 191	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑥)) = -(𝑡 · (log‘𝑥)))
193		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑦 → (log‘𝑤) = (log‘𝑦))
194	193	negeqd 11458	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → -(log‘𝑤) = -(log‘𝑦))
195		negex 11462	. . . . . . . . . . . . . . . 16 ⊢ -(log‘𝑦) ∈ V
196	194, 179, 195	fvmpt 6998	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ⁺ → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦) = -(log‘𝑦))
197	165, 196	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦) = -(log‘𝑦))
198	197	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦)) = ((1 − 𝑡) · -(log‘𝑦)))
199	164	recnd 11246	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (1 − 𝑡) ∈ ℂ)
200	166	recnd 11246	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (log‘𝑦) ∈ ℂ)
201	199, 200	mulneg2d 11672	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · -(log‘𝑦)) = -((1 − 𝑡) · (log‘𝑦)))
202	198, 201	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘𝑦)) = -((1 − 𝑡) · (log‘𝑦)))
203	192, 202	oveq12d 7429	. . . . . . . . . . 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))))
204	161	recnd 11246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → (𝑡 · (log‘𝑥)) ∈ ℂ)
205	167	recnd 11246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑥 < 𝑦) ∧ 𝑡 ∈ (0(,)1)) → ((1 − 𝑡) · (log‘𝑦)) ∈ ℂ)
206	204, 205	negdid 11588	. . . . . . . . . . 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))))
207	203, 206	eqtr4d 2775	. . . . . . . . . 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))))
208	176, 182, 207	3brtr4d 5180	. . . . . . . . 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))))
209	119, 123, 130, 208	scvxcvx 26714	. . . . . . . 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))))
210	119, 123, 130, 5, 137, 10, 153, 209	jensen 26717	. . . . . . 7 ⊢ (𝜑 → (((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))}))) ∈ ℝ⁺ ∧ ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))})))) ≤ ((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))})))))
211	210	simprd 496	. . . . . 6 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))})))) ≤ ((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))}))))
212	131	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (1 / (♯‘𝐴)) ∈ ℝ)
213	139	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 × {(1 / (♯‘𝐴))}) = (𝑘 ∈ 𝐴 ↦ (1 / (♯‘𝐴))))
214	5, 212, 11, 213, 36	offval2 7692	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹) = (𝑘 ∈ 𝐴 ↦ ((1 / (♯‘𝐴)) · (𝐹‘𝑘))))
215	214	oveq2d 7427	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ ((1 / (♯‘𝐴)) · (𝐹‘𝑘)))))
216		cnfldmul 21150	. . . . . . . . . . . 12 ⊢ · = (._r‘ℂ_fld)
217	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ_fld ∈ Ring)
218	109	fmpttd 7116	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)):𝐴⟶ℂ)
219	218, 5, 16	fdmfifsupp 9375	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) finSupp 0)
220	53, 1, 216, 217, 5, 143, 109, 219	gsummulc2 20205	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ ((1 / (♯‘𝐴)) · (𝐹‘𝑘)))) = ((1 / (♯‘𝐴)) · (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))))
221		fss 6734	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐴⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → 𝐹:𝐴⟶ℝ)
222	10, 118, 221	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
223	10, 5, 16	fdmfifsupp 9375	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 finSupp 0)
224	1, 4, 5, 9, 222, 223	gsumsubgcl 19829	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ_fld Σ_g 𝐹) ∈ ℝ)
225	224	recnd 11246	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℂ_fld Σ_g 𝐹) ∈ ℂ)
226	225, 24, 25	divrec2d 11998	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) = ((1 / (♯‘𝐴)) · (ℂ_fld Σ_g 𝐹)))
227	108	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 / (♯‘𝐴)) · (ℂ_fld Σ_g 𝐹)) = ((1 / (♯‘𝐴)) · (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))))
228	226, 227	eqtr2d 2773	. . . . . . . . . . 11 ⊢ (𝜑 → ((1 / (♯‘𝐴)) · (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))) = ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))
229	215, 220, 228	3eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) = ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))
230	229, 152	oveq12d 7429	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))}))) = (((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) / 1))
231	224, 23	nndivred 12270	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) ∈ ℝ)
232	231	recnd 11246	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) ∈ ℂ)
233	232	div1d 11986	. . . . . . . . 9 ⊢ (𝜑 → (((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) / 1) = ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))
234	230, 233	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))}))) = ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))
235	234	fveq2d 6895	. . . . . . 7 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))})))) = ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))
236		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑤 = ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) → (log‘𝑤) = (log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))
237	236	negeqd 11458	. . . . . . . . 9 ⊢ (𝑤 = ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) → -(log‘𝑤) = -(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))
238		negex 11462	. . . . . . . . 9 ⊢ -(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))) ∈ V
239	237, 179, 238	fvmpt 6998	. . . . . . . 8 ⊢ (((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) ∈ ℝ⁺ → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))) = -(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))
240	115, 239	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))) = -(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))
241	235, 240	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤))‘((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · 𝐹)) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))})))) = -(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))
242	53, 1, 216, 217, 5, 143, 31, 17	gsummulc2 20205	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ ((1 / (♯‘𝐴)) · -(log‘(𝐹‘𝑘))))) = ((1 / (♯‘𝐴)) · (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))))
243		negex 11462	. . . . . . . . . . . 12 ⊢ -(log‘(𝐹‘𝑘)) ∈ V
244	243	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -(log‘(𝐹‘𝑘)) ∈ V)
245		eqidd 2733	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) = (𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)))
246		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹‘𝑘) → (log‘𝑤) = (log‘(𝐹‘𝑘)))
247	246	negeqd 11458	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑘) → -(log‘𝑤) = -(log‘(𝐹‘𝑘)))
248	11, 36, 245, 247	fmptco 7129	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))
249	5, 212, 244, 213, 248	offval2 7692	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹)) = (𝑘 ∈ 𝐴 ↦ ((1 / (♯‘𝐴)) · -(log‘(𝐹‘𝑘)))))
250	249	oveq2d 7427	. . . . . . . . 9 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) = (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ ((1 / (♯‘𝐴)) · -(log‘(𝐹‘𝑘))))))
251	19, 24, 25	divrec2d 11998	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) = ((1 / (♯‘𝐴)) · (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘))))))
252	242, 250, 251	3eqtr4d 2782	. . . . . . . 8 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) = ((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)))
253	252, 152	oveq12d 7429	. . . . . . 7 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))}))) = (((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) / 1))
254	117	recnd 11246	. . . . . . . 8 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) ∈ ℂ)
255	254	div1d 11986	. . . . . . 7 ⊢ (𝜑 → (((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) / 1) = ((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)))
256	253, 255	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝐴 × {(1 / (♯‘𝐴))}) ∘_f · ((𝑤 ∈ ℝ⁺ ↦ -(log‘𝑤)) ∘ 𝐹))) / (ℂ_fld Σ_g (𝐴 × {(1 / (♯‘𝐴))}))) = ((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)))
257	211, 241, 256	3brtr3d 5179	. . . . 5 ⊢ (𝜑 → -(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))) ≤ ((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)))
258	116, 117, 257	lenegcon1d 11800	. . . 4 ⊢ (𝜑 → -((ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ -(log‘(𝐹‘𝑘)))) / (♯‘𝐴)) ≤ (log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))
259	107, 258	eqbrtrrd 5172	. . 3 ⊢ (𝜑 → ((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹))) ≤ (log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))
260	131, 104	remulcld 11248	. . . 4 ⊢ (𝜑 → ((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹))) ∈ ℝ)
261		efle 16065	. . . 4 ⊢ ((((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹))) ∈ ℝ ∧ (log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))) ∈ ℝ) → (((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹))) ≤ (log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))) ↔ (exp‘((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹)))) ≤ (exp‘(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))))
262	260, 116, 261	syl2anc 584	. . 3 ⊢ (𝜑 → (((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹))) ≤ (log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))) ↔ (exp‘((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹)))) ≤ (exp‘(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴))))))
263	259, 262	mpbid 231	. 2 ⊢ (𝜑 → (exp‘((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹)))) ≤ (exp‘(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))))
264	100	rpcnd 13022	. . 3 ⊢ (𝜑 → (𝑀 Σ_g 𝐹) ∈ ℂ)
265	100	rpne0d 13025	. . 3 ⊢ (𝜑 → (𝑀 Σ_g 𝐹) ≠ 0)
266	264, 265, 143	cxpefd 26444	. 2 ⊢ (𝜑 → ((𝑀 Σ_g 𝐹)↑_𝑐(1 / (♯‘𝐴))) = (exp‘((1 / (♯‘𝐴)) · (log‘(𝑀 Σ_g 𝐹)))))
267	115	reeflogd 26356	. . 3 ⊢ (𝜑 → (exp‘(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))) = ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))
268	267	eqcomd 2738	. 2 ⊢ (𝜑 → ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)) = (exp‘(log‘((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))))
269	263, 266, 268	3brtr4d 5180	1 ⊢ (𝜑 → ((𝑀 Σ_g 𝐹)↑_𝑐(1 / (♯‘𝐴))) ≤ ((ℂ_fld Σ_g 𝐹) / (♯‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ∘_f cof 7670 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 +∞cpnf 11249 < clt 11252 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 ℕcn 12216 ℤcz 12562 ℝ⁺crp 12978 (,)cioo 13328 [,)cico 13330 [,]cicc 13331 ♯chash 14294 Σcsu 15636 expce 16009 Basecbs 17148 ↾_s cress 17177 0_gc0g 17389 Σ_g cgsu 17390 Mndcmnd 18659 MndHom cmhm 18703 SubMndcsubmnd 18704 ._gcmg 18986 SubGrpcsubg 19036 GrpHom cghm 19127 GrpIso cgim 19171 CMndccmn 19689 Abelcabl 19690 mulGrpcmgp 20028 Ringcrg 20127 CRingccrg 20128 SubRingcsubrg 20457 DivRingcdr 20500 ℂ_fldccnfld 21144 ℝ_fldcrefld 21376 logclog 26287 ↑_𝑐ccxp 26288

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-mulg 18987 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 20434 df-subrg 20459 df-drng 20502 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-refld 21377 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25607 df-dv 25608 df-log 26289 df-cxp 26290

This theorem is referenced by: amgm 26719 amgm2d 43252 amgm3d 43253 amgm4d 43254

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