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Theorem logccv 26055

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 1191	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ⁺)
2	1	rpred 12966	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ)
3		simpl2 1192	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ⁺)
4	3	rpred 12966	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ)
5		simpl3 1193	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 < 𝐵)
6	1	rpgt0d 12969	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 0 < 𝐴)
7	4	ltpnfd 13051	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 < +∞)
8		0xr 11211	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9		pnfxr 11218	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
10		iccssioo 13343	. . . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
13		ioorp 13352	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ⁺
14	12, 13	sseqtrdi 3997	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℝ⁺)
15	14	sselda 3947	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ⁺)
16	15	relogcld 26015	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (log‘𝑥) ∈ ℝ)
17	16	renegcld 11591	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → -(log‘𝑥) ∈ ℝ)
18	17	fmpttd 7068	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)):(𝐴[,]𝐵)⟶ℝ)
19		ax-resscn 11117	. . . . . 6 ⊢ ℝ ⊆ ℂ
20	14	resabs1d 5973	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) = (log ↾ (𝐴[,]𝐵)))
21		ssid 3969	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
22		cncfss 24299	. . . . . . . . . . 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 26030	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
25	23, 24	sselii 3944	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℂ)
26		rescncf 24297	. . . . . . . . 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 2833	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
29		fvres 6866	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((log ↾ (𝐴[,]𝐵))‘𝑥) = (log‘𝑥))
30	29	negeqd 11404	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → -((log ↾ (𝐴[,]𝐵))‘𝑥) = -(log‘𝑥))
31	30	mpteq2ia 5213	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
32	31	eqcomi 2740	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥))
33	32	negfcncf 24323	. . . . . . 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 24298	. . . . . 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 587	. . . . 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 13335	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
39		ltso 11244	. . . . . . . 8 ⊢ < Or ℝ
40		soss 5570	. . . . . . . 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 13360	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
44	43, 14	sstrid 3958	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴(,)𝐵) ⊆ ℝ⁺)
45	44	sselda 3947	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
46	45	rprecred 12977	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
47	46	renegcld 11591	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -(1 / 𝑥) ∈ ℝ)
48	47	fmpttd 7068	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ℝ)
49	48	frnd 6681	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ)
50		soss 5570	. . . . . . . 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 5569	. . . . . . 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 11408	. . . . . . . . 9 ⊢ -(1 / 𝑥) ∈ V
55		eqid 2731	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))
56	54, 55	fnmpti 6649	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Fn (𝐴(,)𝐵)
57		dffn4 6767	. . . . . . . 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 3947	. . . . . . . . . . . 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 12977	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑧) ∈ ℝ)
63	44	sselda 3947	. . . . . . . . . . . 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 12977	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑦) ∈ ℝ)
66	62, 65	ltnegd 11742	. . . . . . . . 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 12984	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (𝑦 < 𝑧 ↔ (1 / 𝑧) < (1 / 𝑦)))
68		oveq2 7370	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
69	68	negeqd 11404	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → -(1 / 𝑥) = -(1 / 𝑦))
70		negex 11408	. . . . . . . . . . . 12 ⊢ -(1 / 𝑦) ∈ V
71	69, 55, 70	fvmpt 6953	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) = -(1 / 𝑦))
72		oveq2 7370	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (1 / 𝑥) = (1 / 𝑧))
73	72	negeqd 11404	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → -(1 / 𝑥) = -(1 / 𝑧))
74		negex 11408	. . . . . . . . . . . 12 ⊢ -(1 / 𝑧) ∈ V
75	73, 55, 74	fvmpt 6953	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧) = -(1 / 𝑧))
76	71, 75	breqan12d 5126	. . . . . . . . . 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 482	. . . . . . . . 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 3193	. . . . . 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 7278	. . . . . 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 11152	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
84	83	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ℝ ∈ {ℝ, ℂ})
85		relogcl 25968	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
86	85	adantl 482	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
87	86	recnd 11192	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
88	87	negcld 11508	. . . . . . 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 7397	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ V)
91		relogf1o 25959	. . . . . . . . . . . . 13 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
92		f1of 6789	. . . . . . . . . . . . 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 6915	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
95		fvres 6866	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
96	95	mpteq2ia 5213	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
97	94, 96	eqtrdi 2787	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
98	97	oveq2d 7378	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
99		dvrelog 26029	. . . . . . . . 9 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
100	98, 99	eqtr3di 2786	. . . . . . . 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 25367	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ -(log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ -(1 / 𝑥)))
102		eqid 2731	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
103	102	tgioo2 24203	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
104		iccntr 24221	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
105	2, 4, 104	syl2anc 584	. . . . . . 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 25363	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
107		isoeq1 7267	. . . . . 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 485	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0(,)1))
111		eqid 2731	. . . 4 ⊢ ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))
112	2, 4, 5, 37, 109, 110, 111	dvcvx 25421	. . 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 11118	. . . . . . . 8 ⊢ 1 ∈ ℂ
114		elioore 13304	. . . . . . . . . 10 ⊢ (𝑇 ∈ (0(,)1) → 𝑇 ∈ ℝ)
115	114	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℝ)
116	115	recnd 11192	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℂ)
117		nncan 11439	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
118	113, 116, 117	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − (1 − 𝑇)) = 𝑇)
119	118	oveq1d 7377	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − (1 − 𝑇)) · 𝐴) = (𝑇 · 𝐴))
120	119	oveq1d 7377	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)))
121		ioossicc 13360	. . . . . . . 8 ⊢ (0(,)1) ⊆ (0[,]1)
122	121, 110	sselid 3945	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0[,]1))
123		iirev 24329	. . . . . . 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 13422	. . . . . 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 1373	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
127	120, 126	eqeltrrd 2833	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
128		fveq2 6847	. . . . . 6 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → (log‘𝑥) = (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
129	128	negeqd 11404	. . . . 5 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → -(log‘𝑥) = -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
130		eqid 2731	. . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
131		negex 11408	. . . . 5 ⊢ -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ V
132	129, 130, 131	fvmpt 6953	. . . 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 12967	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ^*)
135	3	rpxrd 12967	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ^*)
136	2, 4, 5	ltled 11312	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ≤ 𝐵)
137		lbicc2 13391	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138	134, 135, 136, 137	syl3anc 1371	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ (𝐴[,]𝐵))
139		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
140	139	negeqd 11404	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → -(log‘𝑥) = -(log‘𝐴))
141		negex 11408	. . . . . . . . 9 ⊢ -(log‘𝐴) ∈ V
142	140, 130, 141	fvmpt 6953	. . . . . . . 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 7378	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = (𝑇 · -(log‘𝐴)))
145	1	relogcld 26015	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℝ)
146	145	recnd 11192	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℂ)
147	116, 146	mulneg2d 11618	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · -(log‘𝐴)) = -(𝑇 · (log‘𝐴)))
148	144, 147	eqtrd 2771	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = -(𝑇 · (log‘𝐴)))
149		ubicc2 13392	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
150	134, 135, 136, 149	syl3anc 1371	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ (𝐴[,]𝐵))
151		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
152	151	negeqd 11404	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → -(log‘𝑥) = -(log‘𝐵))
153		negex 11408	. . . . . . . . 9 ⊢ -(log‘𝐵) ∈ V
154	152, 130, 153	fvmpt 6953	. . . . . . . 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 7378	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵)) = ((1 − 𝑇) · -(log‘𝐵)))
157		1re 11164	. . . . . . . . 9 ⊢ 1 ∈ ℝ
158		resubcl 11474	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
159	157, 115, 158	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℝ)
160	159	recnd 11192	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℂ)
161	3	relogcld 26015	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℝ)
162	161	recnd 11192	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℂ)
163	160, 162	mulneg2d 11618	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · -(log‘𝐵)) = -((1 − 𝑇) · (log‘𝐵)))
164	156, 163	eqtrd 2771	. . . . 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 7380	. . . 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 11194	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℝ)
167	166	recnd 11192	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℂ)
168	159, 161	remulcld 11194	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℝ)
169	168	recnd 11192	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℂ)
170	167, 169	negdid 11534	. . . 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 2774	. . 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 5141	. 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 11193	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) ∈ ℝ)
174	14, 127	sseldd 3948	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ ℝ⁺)
175	174	relogcld 26015	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ ℝ)
176	173, 175	ltnegd 11742	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3446 ⊆ wss 3913 {cpr 4593 class class class wbr 5110 ↦ cmpt 5193 Po wpo 5548 Or wor 5549 ran crn 5639 ↾ cres 5640 Fn wfn 6496 ⟶wf 6497 –onto→wfo 6499 –1-1-onto→wf1o 6500 ‘cfv 6501 Isom wiso 6502 (class class class)co 7362 ℂcc 11058 ℝcr 11059 0cc0 11060 1c1 11061 + caddc 11063 · cmul 11065 +∞cpnf 11195 ℝ^*cxr 11197 < clt 11198 ≤ cle 11199 − cmin 11394 -cneg 11395 / cdiv 11821 ℝ⁺crp 12924 (,)cioo 13274 [,]cicc 13277 TopOpenctopn 17317 topGenctg 17333 ℂ_fldccnfld 20833 intcnt 22405 –cn→ccncf 24276 D cdv 25264 logclog 25947

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 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9586 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138 ax-addf 11139 ax-mulf 11140

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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9356 df-sup 9387 df-inf 9388 df-oi 9455 df-card 9884 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-z 12509 df-dec 12628 df-uz 12773 df-q 12883 df-rp 12925 df-xneg 13042 df-xadd 13043 df-xmul 13044 df-ioo 13278 df-ioc 13279 df-ico 13280 df-icc 13281 df-fz 13435 df-fzo 13578 df-fl 13707 df-mod 13785 df-seq 13917 df-exp 13978 df-fac 14184 df-bc 14213 df-hash 14241 df-shft 14964 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-limsup 15365 df-clim 15382 df-rlim 15383 df-sum 15583 df-ef 15961 df-sin 15963 df-cos 15964 df-pi 15966 df-struct 17030 df-sets 17047 df-slot 17065 df-ndx 17077 df-base 17095 df-ress 17124 df-plusg 17160 df-mulr 17161 df-starv 17162 df-sca 17163 df-vsca 17164 df-ip 17165 df-tset 17166 df-ple 17167 df-ds 17169 df-unif 17170 df-hom 17171 df-cco 17172 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-submnd 18616 df-mulg 18887 df-cntz 19111 df-cmn 19578 df-psmet 20825 df-xmet 20826 df-met 20827 df-bl 20828 df-mopn 20829 df-fbas 20830 df-fg 20831 df-cnfld 20834 df-top 22280 df-topon 22297 df-topsp 22319 df-bases 22333 df-cld 22407 df-ntr 22408 df-cls 22409 df-nei 22486 df-lp 22524 df-perf 22525 df-cn 22615 df-cnp 22616 df-haus 22703 df-cmp 22775 df-tx 22950 df-hmeo 23143 df-fil 23234 df-fm 23326 df-flim 23327 df-flf 23328 df-xms 23710 df-ms 23711 df-tms 23712 df-cncf 24278 df-limc 25267 df-dv 25268 df-log 25949

This theorem is referenced by: amgmlem 26376 amgmwlem 47369

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