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Theorem logccv 27021

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 1210	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ⁺)
2	1	rpred 13191	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ)
3		simpl2 1211	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ⁺)
4	3	rpred 13191	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ)
5		simpl3 1212	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 < 𝐵)
6	1	rpgt0d 13194	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 0 < 𝐴)
7	4	ltpnfd 13277	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 < +∞)
8		0xr 11383	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9		pnfxr 11390	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
10		iccssioo 13573	. . . . . . . . . . . 12 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 < 𝐴 ∧ 𝐵 < +∞)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
11	8, 9, 10	mpanl12 715	. . . . . . . . . . 11 ⊢ ((0 < 𝐴 ∧ 𝐵 < +∞) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
12	6, 7, 11	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
13		ioorp 13583	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ⁺
14	12, 13	sseqtrdi 3953	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℝ⁺)
15	14	sselda 3913	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ⁺)
16	15	relogcld 26981	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (log‘𝑥) ∈ ℝ)
17	16	renegcld 11770	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → -(log‘𝑥) ∈ ℝ)
18	17	fmpttd 7139	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)):(𝐴[,]𝐵)⟶ℝ)
19		ax-resscn 11284	. . . . . 6 ⊢ ℝ ⊆ ℂ
20	14	resabs1d 6011	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) = (log ↾ (𝐴[,]𝐵)))
21		ssid 3935	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
22		cncfss 25250	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ))
23	19, 21, 22	mp2an 705	. . . . . . . . . 10 ⊢ (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ)
24		relogcn 26996	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
25	23, 24	sselii 3910	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℂ)
26		rescncf 25248	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ ℝ⁺ → ((log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℂ) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
27	14, 25, 26	mpisyl 22	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
28	20, 27	eqeltrrd 2865	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
29		fvres 6927	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((log ↾ (𝐴[,]𝐵))‘𝑥) = (log‘𝑥))
30	29	negeqd 11578	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → -((log ↾ (𝐴[,]𝐵))‘𝑥) = -(log‘𝑥))
31	30	mpteq2ia 5203	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
32	31	eqcomi 2772	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥))
33	32	negfcncf 25274	. . . . . . 7 ⊢ ((log ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
34	28, 33	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
35		cncfcdm 25249	. . . . . 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 599	. . . . 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 260	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
38		ioossre 13565	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
39		ltso 11417	. . . . . . . 8 ⊢ < Or ℝ
40		soss 5583	. . . . . . . 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 13591	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
44	43, 14	sstrid 3924	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴(,)𝐵) ⊆ ℝ⁺)
45	44	sselda 3913	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
46	45	rprecred 13202	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
47	46	renegcld 11770	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -(1 / 𝑥) ∈ ℝ)
48	47	fmpttd 7139	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ℝ)
49	48	frnd 6740	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ)
50		soss 5583	. . . . . . . 8 ⊢ (ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ → ( < Or ℝ → < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))))
51	49, 39, 50	mpisyl 22	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
52		sopo 5582	. . . . . . 7 ⊢ ( < Or ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) → < Po ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
53	51, 52	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → < Po ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
54		negex 11582	. . . . . . . . 9 ⊢ -(1 / 𝑥) ∈ V
55		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))
56	54, 55	fnmpti 6704	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Fn (𝐴(,)𝐵)
57		dffn4 6824	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Fn (𝐴(,)𝐵) ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ontoran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
58	56, 57	mpbi 233	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ontoran (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)⟶ontoran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
60	44	sselda 3913	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → 𝑧 ∈ ℝ⁺)
61	60	adantrl 729	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → 𝑧 ∈ ℝ⁺)
62	61	rprecred 13202	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑧) ∈ ℝ)
63	44	sselda 3913	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
64	63	adantrr 730	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → 𝑦 ∈ ℝ⁺)
65	64	rprecred 13202	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑦) ∈ ℝ)
66	62, 65	ltnegd 11921	. . . . . . . . 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 13209	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (𝑦 < 𝑧 ↔ (1 / 𝑧) < (1 / 𝑦)))
68		oveq2 7451	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
69	68	negeqd 11578	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → -(1 / 𝑥) = -(1 / 𝑦))
70		negex 11582	. . . . . . . . . . . 12 ⊢ -(1 / 𝑦) ∈ V
71	69, 55, 70	fvmpt 7016	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) = -(1 / 𝑦))
72		oveq2 7451	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (1 / 𝑥) = (1 / 𝑧))
73	72	negeqd 11578	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → -(1 / 𝑥) = -(1 / 𝑧))
74		negex 11582	. . . . . . . . . . . 12 ⊢ -(1 / 𝑧) ∈ V
75	73, 55, 74	fvmpt 7016	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧) = -(1 / 𝑧))
76	71, 75	breqan12d 5122	. . . . . . . . . 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 487	. . . . . . . . 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 314	. . . . . . . 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 232	. . . . . . 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 3209	. . . . . 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 7359	. . . . . 6 ⊢ ((( < Or (𝐴(,)𝐵) ∧ < Po ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))) ∧ ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ontoran (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 852	. . . . 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 11319	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
84	83	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ℝ ∈ {ℝ, ℂ})
85		relogcl 26933	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
86	85	adantl 487	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
87	86	recnd 11364	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
88	87	negcld 11683	. . . . . . 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 7478	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ V)
91		relogf1o 26924	. . . . . . . . . . . . 13 ⊢ (log ↾ ℝ⁺):ℝ⁺⟶1-1ontoℝ
92		f1of 6846	. . . . . . . . . . . . 13 ⊢ ((log ↾ ℝ⁺):ℝ⁺⟶1-1ontoℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
93	91, 92	mp1i 14	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
94	93	feqmptd 6976	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
95		fvres 6927	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
96	95	mpteq2ia 5203	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
97	94, 96	eqtrdi 2814	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
98	97	oveq2d 7459	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
99		dvrelog 26995	. . . . . . . . 9 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
100	98, 99	eqtr3di 2813	. . . . . . . 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 26316	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ -(log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ -(1 / 𝑥)))
102		tgioo4 25154	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
103		eqid 2763	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
104		iccntr 25171	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
105	2, 4, 104	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
106	84, 88, 89, 101, 14, 102, 103, 105	dvmptres2 26312	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
107		isoeq1 7348	. . . . . 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 18	. . . . 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 260	. . . 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 490	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0(,)1))
111		eqid 2763	. . . 4 ⊢ ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))
112	2, 4, 5, 37, 109, 110, 111	dvcvx 26370	. . 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 11285	. . . . . . . 8 ⊢ 1 ∈ ℂ
114		elioore 13533	. . . . . . . . . 10 ⊢ (𝑇 ∈ (0(,)1) → 𝑇 ∈ ℝ)
115	114	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℝ)
116	115	recnd 11364	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℂ)
117		nncan 11614	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
118	113, 116, 117	sylancr 599	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − (1 − 𝑇)) = 𝑇)
119	118	oveq1d 7458	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − (1 − 𝑇)) · 𝐴) = (𝑇 · 𝐴))
120	119	oveq1d 7458	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)))
121		ioossicc 13591	. . . . . . . 8 ⊢ (0(,)1) ⊆ (0[,]1)
122	121, 110	sselid 3911	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0[,]1))
123		iirev 25280	. . . . . . 7 ⊢ (𝑇 ∈ (0[,]1) → (1 − 𝑇) ∈ (0[,]1))
124	122, 123	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ (0[,]1))
125		lincmb01cmp 13653	. . . . . 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 1401	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
127	120, 126	eqeltrrd 2865	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
128		fveq2 6908	. . . . . 6 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → (log‘𝑥) = (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
129	128	negeqd 11578	. . . . 5 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → -(log‘𝑥) = -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
130		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
131		negex 11582	. . . . 5 ⊢ -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ V
132	129, 130, 131	fvmpt 7016	. . . 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 18	. . 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 13192	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ^*)
135	3	rpxrd 13192	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ^*)
136	2, 4, 5	ltled 11485	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ≤ 𝐵)
137		lbicc2 13622	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138	134, 135, 136, 137	syl3anc 1399	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ (𝐴[,]𝐵))
139		fveq2 6908	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
140	139	negeqd 11578	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → -(log‘𝑥) = -(log‘𝐴))
141		negex 11582	. . . . . . . . 9 ⊢ -(log‘𝐴) ∈ V
142	140, 130, 141	fvmpt 7016	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴) = -(log‘𝐴))
143	138, 142	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴) = -(log‘𝐴))
144	143	oveq2d 7459	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = (𝑇 · -(log‘𝐴)))
145	1	relogcld 26981	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℝ)
146	145	recnd 11364	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℂ)
147	116, 146	mulneg2d 11797	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · -(log‘𝐴)) = -(𝑇 · (log‘𝐴)))
148	144, 147	eqtrd 2798	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = -(𝑇 · (log‘𝐴)))
149		ubicc2 13623	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
150	134, 135, 136, 149	syl3anc 1399	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ (𝐴[,]𝐵))
151		fveq2 6908	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
152	151	negeqd 11578	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → -(log‘𝑥) = -(log‘𝐵))
153		negex 11582	. . . . . . . . 9 ⊢ -(log‘𝐵) ∈ V
154	152, 130, 153	fvmpt 7016	. . . . . . . 8 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵) = -(log‘𝐵))
155	150, 154	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵) = -(log‘𝐵))
156	155	oveq2d 7459	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵)) = ((1 − 𝑇) · -(log‘𝐵)))
157		1re 11335	. . . . . . . . 9 ⊢ 1 ∈ ℝ
158		resubcl 11649	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
159	157, 115, 158	sylancr 599	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℝ)
160	159	recnd 11364	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℂ)
161	3	relogcld 26981	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℝ)
162	161	recnd 11364	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℂ)
163	160, 162	mulneg2d 11797	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · -(log‘𝐵)) = -((1 − 𝑇) · (log‘𝐵)))
164	156, 163	eqtrd 2798	. . . . 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 7461	. . . 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 11366	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℝ)
167	166	recnd 11364	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℂ)
168	159, 161	remulcld 11366	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℝ)
169	168	recnd 11364	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℂ)
170	167, 169	negdid 11709	. . . 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 2801	. . 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 5139	. 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 11365	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) ∈ ℝ)
174	14, 127	sseldd 3914	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ ℝ⁺)
175	174	relogcld 26981	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ ℝ)
176	173, 175	ltnegd 11921	. 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 260	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 {cpr 4590 class class class wbr 5106 ↦ cmpt 5189 Po wpo 5561 Or wor 5562 ran crn 5656 ↾ cres 5657 Fn wfn 6553 ⟶wf 6554 ⟶ontowfo 6556 ⟶1-1ontowf1o 6557 ‘cfv 6558 Isom wiso 6559 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 1c1 11228 + caddc 11230 · cmul 11232 +∞cpnf 11367 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371 − cmin 11568 -cneg 11569 / cdiv 12000 ℝ⁺crp 13147 (,)cioo 13503 [,]cicc 13506 TopOpenctopn 17621 topGenctg 17637 ℂ_fldccnfld 21708 intcnt 23365 –cn→ccncf 25227 D cdv 26213 logclog 26912

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ioo 13507 df-ioc 13508 df-ico 13509 df-icc 13510 df-fz 13667 df-fzo 13816 df-fl 13959 df-mod 14037 df-seq 14172 df-exp 14232 df-fac 14445 df-bc 14474 df-hash 14502 df-shft 15249 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-limsup 15667 df-clim 15684 df-rlim 15685 df-sum 15883 df-ef 16262 df-sin 16264 df-cos 16265 df-pi 16267 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-rest 17622 df-topn 17623 df-0g 17641 df-gsum 17642 df-topgen 17643 df-pt 17644 df-prds 17647 df-xrs 17703 df-qtop 17708 df-imas 17709 df-xps 17711 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-submnd 19008 df-mulg 19307 df-cntz 19561 df-cmn 20026 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-fbas 21705 df-fg 21706 df-cnfld 21709 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-cld 23367 df-ntr 23368 df-cls 23369 df-nei 23446 df-lp 23484 df-perf 23485 df-cn 23575 df-cnp 23576 df-haus 23663 df-cmp 23735 df-tx 23911 df-hmeo 24104 df-fil 24195 df-fm 24287 df-flim 24288 df-flf 24289 df-xms 24669 df-ms 24670 df-tms 24671 df-cncf 25229 df-limc 26216 df-dv 26217 df-log 26914

This theorem is used by: amgmlem 27347 amgmwlem 51267

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