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Theorem logccv 26627

Description: The natural logarithm function on the reals is a strictly concave function. (Contributed by Mario Carneiro, 20-Jun-2015.)

**Assertion**
Ref	Expression
logccv	⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) < (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))

Proof of Theorem logccv Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1188	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ⁺)
2	1	rpred 13048	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ)
3		simpl2 1189	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ⁺)
4	3	rpred 13048	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ)
5		simpl3 1190	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 < 𝐵)
6	1	rpgt0d 13051	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 0 < 𝐴)
7	4	ltpnfd 13133	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 < +∞)
8		0xr 11291	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9		pnfxr 11298	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
10		iccssioo 13425	. . . . . . . . . . . 12 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 < 𝐴 ∧ 𝐵 < +∞)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
11	8, 9, 10	mpanl12 700	. . . . . . . . . . 11 ⊢ ((0 < 𝐴 ∧ 𝐵 < +∞) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
12	6, 7, 11	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
13		ioorp 13434	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ⁺
14	12, 13	sseqtrdi 4028	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℝ⁺)
15	14	sselda 3977	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ⁺)
16	15	relogcld 26587	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (log‘𝑥) ∈ ℝ)
17	16	renegcld 11671	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → -(log‘𝑥) ∈ ℝ)
18	17	fmpttd 7122	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)):(𝐴[,]𝐵)⟶ℝ)
19		ax-resscn 11195	. . . . . 6 ⊢ ℝ ⊆ ℂ
20	14	resabs1d 6012	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) = (log ↾ (𝐴[,]𝐵)))
21		ssid 4000	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
22		cncfss 24849	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ))
23	19, 21, 22	mp2an 690	. . . . . . . . . 10 ⊢ (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ)
24		relogcn 26602	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
25	23, 24	sselii 3974	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℂ)
26		rescncf 24847	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ ℝ⁺ → ((log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℂ) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
27	14, 25, 26	mpisyl 21	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
28	20, 27	eqeltrrd 2826	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
29		fvres 6913	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((log ↾ (𝐴[,]𝐵))‘𝑥) = (log‘𝑥))
30	29	negeqd 11484	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → -((log ↾ (𝐴[,]𝐵))‘𝑥) = -(log‘𝑥))
31	30	mpteq2ia 5251	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
32	31	eqcomi 2734	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥))
33	32	negfcncf 24874	. . . . . . 7 ⊢ ((log ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
34	28, 33	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
35		cncfcdm 24848	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)):(𝐴[,]𝐵)⟶ℝ))
36	19, 34, 35	sylancr 585	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)):(𝐴[,]𝐵)⟶ℝ))
37	18, 36	mpbird 256	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
38		ioossre 13417	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
39		ltso 11324	. . . . . . . 8 ⊢ < Or ℝ
40		soss 5609	. . . . . . . 8 ⊢ ((𝐴(,)𝐵) ⊆ ℝ → ( < Or ℝ → < Or (𝐴(,)𝐵)))
41	38, 39, 40	mp2 9	. . . . . . 7 ⊢ < Or (𝐴(,)𝐵)
42	41	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → < Or (𝐴(,)𝐵))
43		ioossicc 13442	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
44	43, 14	sstrid 3989	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴(,)𝐵) ⊆ ℝ⁺)
45	44	sselda 3977	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
46	45	rprecred 13059	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
47	46	renegcld 11671	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -(1 / 𝑥) ∈ ℝ)
48	47	fmpttd 7122	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ℝ)
49	48	frnd 6729	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ)
50		soss 5609	. . . . . . . 8 ⊢ (ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ → ( < Or ℝ → < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))))
51	49, 39, 50	mpisyl 21	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
52		sopo 5608	. . . . . . 7 ⊢ ( < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) → < Po ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
53	51, 52	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → < Po ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
54		negex 11488	. . . . . . . . 9 ⊢ -(1 / 𝑥) ∈ V
55		eqid 2725	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))
56	54, 55	fnmpti 6697	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Fn (𝐴(,)𝐵)
57		dffn4 6814	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Fn (𝐴(,)𝐵) ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)–onto→ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
58	56, 57	mpbi 229	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)–onto→ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))
59	58	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)–onto→ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
60	44	sselda 3977	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → 𝑧 ∈ ℝ⁺)
61	60	adantrl 714	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → 𝑧 ∈ ℝ⁺)
62	61	rprecred 13059	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑧) ∈ ℝ)
63	44	sselda 3977	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
64	63	adantrr 715	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → 𝑦 ∈ ℝ⁺)
65	64	rprecred 13059	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑦) ∈ ℝ)
66	62, 65	ltnegd 11822	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → ((1 / 𝑧) < (1 / 𝑦) ↔ -(1 / 𝑦) < -(1 / 𝑧)))
67	64, 61	ltrecd 13066	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (𝑦 < 𝑧 ↔ (1 / 𝑧) < (1 / 𝑦)))
68		oveq2 7425	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
69	68	negeqd 11484	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → -(1 / 𝑥) = -(1 / 𝑦))
70		negex 11488	. . . . . . . . . . . 12 ⊢ -(1 / 𝑦) ∈ V
71	69, 55, 70	fvmpt 7002	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) = -(1 / 𝑦))
72		oveq2 7425	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (1 / 𝑥) = (1 / 𝑧))
73	72	negeqd 11484	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → -(1 / 𝑥) = -(1 / 𝑧))
74		negex 11488	. . . . . . . . . . . 12 ⊢ -(1 / 𝑧) ∈ V
75	73, 55, 74	fvmpt 7002	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧) = -(1 / 𝑧))
76	71, 75	breqan12d 5164	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) < ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧) ↔ -(1 / 𝑦) < -(1 / 𝑧)))
77	76	adantl 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) < ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧) ↔ -(1 / 𝑦) < -(1 / 𝑧)))
78	66, 67, 77	3bitr4d 310	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (𝑦 < 𝑧 ↔ ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) < ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧)))
79	78	biimpd 228	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (𝑦 < 𝑧 → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) < ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧)))
80	79	ralrimivva 3191	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ∀𝑦 ∈ (𝐴(,)𝐵)∀𝑧 ∈ (𝐴(,)𝐵)(𝑦 < 𝑧 → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) < ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧)))
81		soisoi 7333	. . . . . 6 ⊢ ((( < Or (𝐴(,)𝐵) ∧ < Po ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))) ∧ ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)–onto→ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ∧ ∀𝑦 ∈ (𝐴(,)𝐵)∀𝑧 ∈ (𝐴(,)𝐵)(𝑦 < 𝑧 → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) < ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧)))) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Isom < , < ((𝐴(,)𝐵), ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))))
82	42, 53, 59, 80, 81	syl22anc 837	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Isom < , < ((𝐴(,)𝐵), ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))))
83		reelprrecn 11230	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
84	83	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ℝ ∈ {ℝ, ℂ})
85		relogcl 26539	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
86	85	adantl 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
87	86	recnd 11272	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
88	87	negcld 11588	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → -(log‘𝑥) ∈ ℂ)
89	54	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → -(1 / 𝑥) ∈ V)
90		ovexd 7452	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ V)
91		relogf1o 26530	. . . . . . . . . . . . 13 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
92		f1of 6836	. . . . . . . . . . . . 13 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
93	91, 92	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
94	93	feqmptd 6964	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
95		fvres 6913	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
96	95	mpteq2ia 5251	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
97	94, 96	eqtrdi 2781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
98	97	oveq2d 7433	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
99		dvrelog 26601	. . . . . . . . 9 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
100	98, 99	eqtr3di 2780	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
101	84, 87, 90, 100	dvmptneg 25928	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ -(log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ -(1 / 𝑥)))
102		eqid 2725	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
103	102	tgioo2 24749	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
104		iccntr 24767	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
105	2, 4, 104	syl2anc 582	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
106	84, 88, 89, 101, 14, 103, 102, 105	dvmptres2 25924	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
107		isoeq1 7322	. . . . . 6 ⊢ ((ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) → ((ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) Isom < , < ((𝐴(,)𝐵), ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))) ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Isom < , < ((𝐴(,)𝐵), ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))))
108	106, 107	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) Isom < , < ((𝐴(,)𝐵), ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))) ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Isom < , < ((𝐴(,)𝐵), ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))))
109	82, 108	mpbird 256	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) Isom < , < ((𝐴(,)𝐵), ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))))
110		simpr 483	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0(,)1))
111		eqid 2725	. . . 4 ⊢ ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))
112	2, 4, 5, 37, 109, 110, 111	dvcvx 25983	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) < ((𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) + ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵))))
113		ax-1cn 11196	. . . . . . . 8 ⊢ 1 ∈ ℂ
114		elioore 13386	. . . . . . . . . 10 ⊢ (𝑇 ∈ (0(,)1) → 𝑇 ∈ ℝ)
115	114	adantl 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℝ)
116	115	recnd 11272	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℂ)
117		nncan 11519	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
118	113, 116, 117	sylancr 585	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − (1 − 𝑇)) = 𝑇)
119	118	oveq1d 7432	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − (1 − 𝑇)) · 𝐴) = (𝑇 · 𝐴))
120	119	oveq1d 7432	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)))
121		ioossicc 13442	. . . . . . . 8 ⊢ (0(,)1) ⊆ (0[,]1)
122	121, 110	sselid 3975	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0[,]1))
123		iirev 24880	. . . . . . 7 ⊢ (𝑇 ∈ (0[,]1) → (1 − 𝑇) ∈ (0[,]1))
124	122, 123	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ (0[,]1))
125		lincmb01cmp 13504	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ (1 − 𝑇) ∈ (0[,]1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
126	2, 4, 5, 124, 125	syl31anc 1370	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
127	120, 126	eqeltrrd 2826	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
128		fveq2 6894	. . . . . 6 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → (log‘𝑥) = (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
129	128	negeqd 11484	. . . . 5 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → -(log‘𝑥) = -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
130		eqid 2725	. . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
131		negex 11488	. . . . 5 ⊢ -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ V
132	129, 130, 131	fvmpt 7002	. . . 4 ⊢ (((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) = -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
133	127, 132	syl 17	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) = -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
134	1	rpxrd 13049	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ^*)
135	3	rpxrd 13049	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ^*)
136	2, 4, 5	ltled 11392	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ≤ 𝐵)
137		lbicc2 13473	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138	134, 135, 136, 137	syl3anc 1368	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ (𝐴[,]𝐵))
139		fveq2 6894	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
140	139	negeqd 11484	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → -(log‘𝑥) = -(log‘𝐴))
141		negex 11488	. . . . . . . . 9 ⊢ -(log‘𝐴) ∈ V
142	140, 130, 141	fvmpt 7002	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴) = -(log‘𝐴))
143	138, 142	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴) = -(log‘𝐴))
144	143	oveq2d 7433	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = (𝑇 · -(log‘𝐴)))
145	1	relogcld 26587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℝ)
146	145	recnd 11272	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℂ)
147	116, 146	mulneg2d 11698	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · -(log‘𝐴)) = -(𝑇 · (log‘𝐴)))
148	144, 147	eqtrd 2765	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = -(𝑇 · (log‘𝐴)))
149		ubicc2 13474	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
150	134, 135, 136, 149	syl3anc 1368	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ (𝐴[,]𝐵))
151		fveq2 6894	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
152	151	negeqd 11484	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → -(log‘𝑥) = -(log‘𝐵))
153		negex 11488	. . . . . . . . 9 ⊢ -(log‘𝐵) ∈ V
154	152, 130, 153	fvmpt 7002	. . . . . . . 8 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵) = -(log‘𝐵))
155	150, 154	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵) = -(log‘𝐵))
156	155	oveq2d 7433	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵)) = ((1 − 𝑇) · -(log‘𝐵)))
157		1re 11244	. . . . . . . . 9 ⊢ 1 ∈ ℝ
158		resubcl 11554	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
159	157, 115, 158	sylancr 585	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℝ)
160	159	recnd 11272	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℂ)
161	3	relogcld 26587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℝ)
162	161	recnd 11272	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℂ)
163	160, 162	mulneg2d 11698	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · -(log‘𝐵)) = -((1 − 𝑇) · (log‘𝐵)))
164	156, 163	eqtrd 2765	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵)) = -((1 − 𝑇) · (log‘𝐵)))
165	148, 164	oveq12d 7435	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) + ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵))) = (-(𝑇 · (log‘𝐴)) + -((1 − 𝑇) · (log‘𝐵))))
166	115, 145	remulcld 11274	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℝ)
167	166	recnd 11272	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℂ)
168	159, 161	remulcld 11274	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℝ)
169	168	recnd 11272	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℂ)
170	167, 169	negdid 11614	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → -((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) = (-(𝑇 · (log‘𝐴)) + -((1 − 𝑇) · (log‘𝐵))))
171	165, 170	eqtr4d 2768	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) + ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵))) = -((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))))
172	112, 133, 171	3brtr3d 5179	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) < -((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))))
173	166, 168	readdcld 11273	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) ∈ ℝ)
174	14, 127	sseldd 3978	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ ℝ⁺)
175	174	relogcld 26587	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ ℝ)
176	173, 175	ltnegd 11822	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) < (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ↔ -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) < -((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵)))))
177	172, 176	mpbird 256	1 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) < (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ⊆ wss 3945 {cpr 4631 class class class wbr 5148 ↦ cmpt 5231 Po wpo 5587 Or wor 5588 ran crn 5678 ↾ cres 5679 Fn wfn 6542 ⟶wf 6543 –onto→wfo 6545 –1-1-onto→wf1o 6546 ‘cfv 6547 Isom wiso 6548 (class class class)co 7417 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 +∞cpnf 11275 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279 − cmin 11474 -cneg 11475 / cdiv 11901 ℝ⁺crp 13006 (,)cioo 13356 [,]cicc 13359 TopOpenctopn 17402 topGenctg 17418 ℂ_fldccnfld 21283 intcnt 22951 –cn→ccncf 24826 D cdv 25822 logclog 26518

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 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-sin 16045 df-cos 16046 df-pi 16048 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22826 df-topon 22843 df-topsp 22865 df-bases 22879 df-cld 22953 df-ntr 22954 df-cls 22955 df-nei 23032 df-lp 23070 df-perf 23071 df-cn 23161 df-cnp 23162 df-haus 23249 df-cmp 23321 df-tx 23496 df-hmeo 23689 df-fil 23780 df-fm 23872 df-flim 23873 df-flf 23874 df-xms 24256 df-ms 24257 df-tms 24258 df-cncf 24828 df-limc 25825 df-dv 25826 df-log 26520

This theorem is referenced by: amgmlem 26952 amgmwlem 48347

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