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Theorem logccv 24454

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 1084	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ⁺)
2	1	rpred 11910	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ)
3		simpl2 1085	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ⁺)
4	3	rpred 11910	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ)
5		simpl3 1086	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 < 𝐵)
6	1	rpgt0d 11913	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 0 < 𝐴)
7		ltpnf 11992	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
8	4, 7	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 < +∞)
9		0xr 10124	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
10		pnfxr 10130	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
11		iccssioo 12280	. . . . . . . . . . . 12 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 < 𝐴 ∧ 𝐵 < +∞)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
12	9, 10, 11	mpanl12 718	. . . . . . . . . . 11 ⊢ ((0 < 𝐴 ∧ 𝐵 < +∞) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
13	6, 8, 12	syl2anc 694	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
14		ioorp 12289	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ⁺
15	13, 14	syl6sseq 3684	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℝ⁺)
16	15	sselda 3636	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ⁺)
17	16	relogcld 24414	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (log‘𝑥) ∈ ℝ)
18	17	renegcld 10495	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → -(log‘𝑥) ∈ ℝ)
19		eqid 2651	. . . . . 6 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
20	18, 19	fmptd 6425	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)):(𝐴[,]𝐵)⟶ℝ)
21		ax-resscn 10031	. . . . . 6 ⊢ ℝ ⊆ ℂ
22	15	resabs1d 5463	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) = (log ↾ (𝐴[,]𝐵)))
23		ssid 3657	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
24		cncfss 22749	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ))
25	21, 23, 24	mp2an 708	. . . . . . . . . 10 ⊢ (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ)
26		relogcn 24429	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
27	25, 26	sselii 3633	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℂ)
28		rescncf 22747	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ ℝ⁺ → ((log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℂ) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
29	15, 27, 28	mpisyl 21	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
30	22, 29	eqeltrrd 2731	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
31		fvres 6245	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((log ↾ (𝐴[,]𝐵))‘𝑥) = (log‘𝑥))
32	31	negeqd 10313	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → -((log ↾ (𝐴[,]𝐵))‘𝑥) = -(log‘𝑥))
33	32	mpteq2ia 4773	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
34	33	eqcomi 2660	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥))
35	34	negfcncf 22769	. . . . . . 7 ⊢ ((log ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
36	30, 35	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
37		cncffvrn 22748	. . . . . 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))
38	21, 36, 37	sylancr 696	. . . . 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))
39	20, 38	mpbird 247	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
40		ioossre 12273	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
41		ltso 10156	. . . . . . . 8 ⊢ < Or ℝ
42		soss 5082	. . . . . . . 8 ⊢ ((𝐴(,)𝐵) ⊆ ℝ → ( < Or ℝ → < Or (𝐴(,)𝐵)))
43	40, 41, 42	mp2 9	. . . . . . 7 ⊢ < Or (𝐴(,)𝐵)
44	43	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → < Or (𝐴(,)𝐵))
45		ioossicc 12297	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
46	45, 15	syl5ss 3647	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴(,)𝐵) ⊆ ℝ⁺)
47	46	sselda 3636	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
48	47	rprecred 11921	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
49	48	renegcld 10495	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -(1 / 𝑥) ∈ ℝ)
50		eqid 2651	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))
51	49, 50	fmptd 6425	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ℝ)
52		frn 6091	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ℝ → ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ)
53	51, 52	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ)
54		soss 5082	. . . . . . . 8 ⊢ (ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ → ( < Or ℝ → < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))))
55	53, 41, 54	mpisyl 21	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
56		sopo 5081	. . . . . . 7 ⊢ ( < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) → < Po ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
57	55, 56	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → < Po ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
58		negex 10317	. . . . . . . . 9 ⊢ -(1 / 𝑥) ∈ V
59	58, 50	fnmpti 6060	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Fn (𝐴(,)𝐵)
60		dffn4 6159	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Fn (𝐴(,)𝐵) ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)–onto→ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
61	59, 60	mpbi 220	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)–onto→ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))
62	61	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)))
63	46	sselda 3636	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → 𝑧 ∈ ℝ⁺)
64	63	adantrl 752	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → 𝑧 ∈ ℝ⁺)
65	64	rprecred 11921	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑧) ∈ ℝ)
66	46	sselda 3636	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
67	66	adantrr 753	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → 𝑦 ∈ ℝ⁺)
68	67	rprecred 11921	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑦) ∈ ℝ)
69	65, 68	ltnegd 10643	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → ((1 / 𝑧) < (1 / 𝑦) ↔ -(1 / 𝑦) < -(1 / 𝑧)))
70	67, 64	ltrecd 11928	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (𝑦 < 𝑧 ↔ (1 / 𝑧) < (1 / 𝑦)))
71		oveq2 6698	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
72	71	negeqd 10313	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → -(1 / 𝑥) = -(1 / 𝑦))
73		negex 10317	. . . . . . . . . . . 12 ⊢ -(1 / 𝑦) ∈ V
74	72, 50, 73	fvmpt 6321	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) = -(1 / 𝑦))
75		oveq2 6698	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (1 / 𝑥) = (1 / 𝑧))
76	75	negeqd 10313	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → -(1 / 𝑥) = -(1 / 𝑧))
77		negex 10317	. . . . . . . . . . . 12 ⊢ -(1 / 𝑧) ∈ V
78	76, 50, 77	fvmpt 6321	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧) = -(1 / 𝑧))
79	74, 78	breqan12d 4701	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) < ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧) ↔ -(1 / 𝑦) < -(1 / 𝑧)))
80	79	adantl 481	. . . . . . . . 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)))
81	69, 70, 80	3bitr4d 300	. . . . . . . 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)))
82	81	biimpd 219	. . . . . . 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)))
83	82	ralrimivva 3000	. . . . . 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)))
84		soisoi 6618	. . . . . 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))))
85	44, 57, 62, 83, 84	syl22anc 1367	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Isom < , < ((𝐴(,)𝐵), ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))))
86		reelprrecn 10066	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
87	86	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ℝ ∈ {ℝ, ℂ})
88		relogcl 24367	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
89	88	adantl 481	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
90	89	recnd 10106	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
91	90	negcld 10417	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → -(log‘𝑥) ∈ ℂ)
92	58	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → -(1 / 𝑥) ∈ V)
93		ovexd 6720	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ V)
94		dvrelog 24428	. . . . . . . . 9 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
95		relogf1o 24358	. . . . . . . . . . . . 13 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
96		f1of 6175	. . . . . . . . . . . . 13 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
97	95, 96	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
98	97	feqmptd 6288	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
99		fvres 6245	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
100	99	mpteq2ia 4773	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
101	98, 100	syl6eq 2701	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
102	101	oveq2d 6706	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
103	94, 102	syl5reqr 2700	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
104	87, 90, 93, 103	dvmptneg 23774	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ -(log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ -(1 / 𝑥)))
105		eqid 2651	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
106	105	tgioo2 22653	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
107		iccntr 22671	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
108	2, 4, 107	syl2anc 694	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
109	87, 91, 92, 104, 15, 106, 105, 108	dvmptres2 23770	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
110		isoeq1 6607	. . . . . 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)))))
111	109, 110	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)))))
112	85, 111	mpbird 247	. . . 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))))
113		simpr 476	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0(,)1))
114		eqid 2651	. . . 4 ⊢ ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))
115	2, 4, 5, 39, 112, 113, 114	dvcvx 23828	. . 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))))
116		ax-1cn 10032	. . . . . . . 8 ⊢ 1 ∈ ℂ
117		elioore 12243	. . . . . . . . . 10 ⊢ (𝑇 ∈ (0(,)1) → 𝑇 ∈ ℝ)
118	117	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℝ)
119	118	recnd 10106	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℂ)
120		nncan 10348	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
121	116, 119, 120	sylancr 696	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − (1 − 𝑇)) = 𝑇)
122	121	oveq1d 6705	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − (1 − 𝑇)) · 𝐴) = (𝑇 · 𝐴))
123	122	oveq1d 6705	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)))
124		ioossicc 12297	. . . . . . . 8 ⊢ (0(,)1) ⊆ (0[,]1)
125	124, 113	sseldi 3634	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0[,]1))
126		iirev 22775	. . . . . . 7 ⊢ (𝑇 ∈ (0[,]1) → (1 − 𝑇) ∈ (0[,]1))
127	125, 126	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ (0[,]1))
128		lincmb01cmp 12353	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ (1 − 𝑇) ∈ (0[,]1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
129	2, 4, 5, 127, 128	syl31anc 1369	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
130	123, 129	eqeltrrd 2731	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
131		fveq2 6229	. . . . . 6 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → (log‘𝑥) = (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
132	131	negeqd 10313	. . . . 5 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → -(log‘𝑥) = -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
133		negex 10317	. . . . 5 ⊢ -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ V
134	132, 19, 133	fvmpt 6321	. . . 4 ⊢ (((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) = -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
135	130, 134	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))))
136	1	rpxrd 11911	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ^*)
137	3	rpxrd 11911	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ^*)
138	2, 4, 5	ltled 10223	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ≤ 𝐵)
139		lbicc2 12326	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
140	136, 137, 138, 139	syl3anc 1366	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ (𝐴[,]𝐵))
141		fveq2 6229	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
142	141	negeqd 10313	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → -(log‘𝑥) = -(log‘𝐴))
143		negex 10317	. . . . . . . . 9 ⊢ -(log‘𝐴) ∈ V
144	142, 19, 143	fvmpt 6321	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴) = -(log‘𝐴))
145	140, 144	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴) = -(log‘𝐴))
146	145	oveq2d 6706	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = (𝑇 · -(log‘𝐴)))
147	1	relogcld 24414	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℝ)
148	147	recnd 10106	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℂ)
149	119, 148	mulneg2d 10522	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · -(log‘𝐴)) = -(𝑇 · (log‘𝐴)))
150	146, 149	eqtrd 2685	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = -(𝑇 · (log‘𝐴)))
151		ubicc2 12327	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
152	136, 137, 138, 151	syl3anc 1366	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ (𝐴[,]𝐵))
153		fveq2 6229	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
154	153	negeqd 10313	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → -(log‘𝑥) = -(log‘𝐵))
155		negex 10317	. . . . . . . . 9 ⊢ -(log‘𝐵) ∈ V
156	154, 19, 155	fvmpt 6321	. . . . . . . 8 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵) = -(log‘𝐵))
157	152, 156	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵) = -(log‘𝐵))
158	157	oveq2d 6706	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵)) = ((1 − 𝑇) · -(log‘𝐵)))
159		1re 10077	. . . . . . . . 9 ⊢ 1 ∈ ℝ
160		resubcl 10383	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
161	159, 118, 160	sylancr 696	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℝ)
162	161	recnd 10106	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℂ)
163	3	relogcld 24414	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℝ)
164	163	recnd 10106	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℂ)
165	162, 164	mulneg2d 10522	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · -(log‘𝐵)) = -((1 − 𝑇) · (log‘𝐵)))
166	158, 165	eqtrd 2685	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵)) = -((1 − 𝑇) · (log‘𝐵)))
167	150, 166	oveq12d 6708	. . . 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))))
168	118, 147	remulcld 10108	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℝ)
169	168	recnd 10106	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℂ)
170	161, 163	remulcld 10108	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℝ)
171	170	recnd 10106	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℂ)
172	169, 171	negdid 10443	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → -((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) = (-(𝑇 · (log‘𝐴)) + -((1 − 𝑇) · (log‘𝐵))))
173	167, 172	eqtr4d 2688	. . 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))))
174	115, 135, 173	3brtr3d 4716	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) < -((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))))
175	168, 170	readdcld 10107	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) ∈ ℝ)
176	15, 130	sseldd 3637	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ ℝ⁺)
177	176	relogcld 24414	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ ℝ)
178	175, 177	ltnegd 10643	. 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)))))
179	174, 178	mpbird 247	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 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 ⊆ wss 3607 {cpr 4212 class class class wbr 4685 ↦ cmpt 4762 Po wpo 5062 Or wor 5063 ran crn 5144 ↾ cres 5145 Fn wfn 5921 ⟶wf 5922 –onto→wfo 5924 –1-1-onto→wf1o 5925 ‘cfv 5926 Isom wiso 5927 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 +∞cpnf 10109 ℝ^*cxr 10111 < clt 10112 ≤ cle 10113 − cmin 10304 -cneg 10305 / cdiv 10722 ℝ⁺crp 11870 (,)cioo 12213 [,]cicc 12216 TopOpenctopn 16129 topGenctg 16145 ℂ_fldccnfld 19794 intcnt 20869 –cn→ccncf 22726 D cdv 23672 logclog 24346

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-pi 14847 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-cmp 21238 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-log 24348

This theorem is referenced by: amgmlem 24761 amgmwlem 42876

W3C validator