logccv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > logccv		Structured version Visualization version GIF version

Theorem logccv 26612

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 1193	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ⁺)
2	1	rpred 12950	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ)
3		simpl2 1194	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ⁺)
4	3	rpred 12950	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ)
5		simpl3 1195	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 < 𝐵)
6	1	rpgt0d 12953	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 0 < 𝐴)
7	4	ltpnfd 13036	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 < +∞)
8		0xr 11180	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9		pnfxr 11187	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
10		iccssioo 13332	. . . . . . . . . . . 12 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 < 𝐴 ∧ 𝐵 < +∞)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
11	8, 9, 10	mpanl12 703	. . . . . . . . . . 11 ⊢ ((0 < 𝐴 ∧ 𝐵 < +∞) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
12	6, 7, 11	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ (0(,)+∞))
13		ioorp 13342	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ⁺
14	12, 13	sseqtrdi 3963	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℝ⁺)
15	14	sselda 3922	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ⁺)
16	15	relogcld 26572	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (log‘𝑥) ∈ ℝ)
17	16	renegcld 11565	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → -(log‘𝑥) ∈ ℝ)
18	17	fmpttd 7059	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)):(𝐴[,]𝐵)⟶ℝ)
19		ax-resscn 11084	. . . . . 6 ⊢ ℝ ⊆ ℂ
20	14	resabs1d 5965	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((log ↾ ℝ⁺) ↾ (𝐴[,]𝐵)) = (log ↾ (𝐴[,]𝐵)))
21		ssid 3945	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
22		cncfss 24844	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ))
23	19, 21, 22	mp2an 693	. . . . . . . . . 10 ⊢ (ℝ⁺–cn→ℝ) ⊆ (ℝ⁺–cn→ℂ)
24		relogcn 26587	. . . . . . . . . 10 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
25	23, 24	sselii 3919	. . . . . . . . 9 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℂ)
26		rescncf 24842	. . . . . . . . 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 2838	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
29		fvres 6851	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((log ↾ (𝐴[,]𝐵))‘𝑥) = (log‘𝑥))
30	29	negeqd 11375	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → -((log ↾ (𝐴[,]𝐵))‘𝑥) = -(log‘𝑥))
31	30	mpteq2ia 5181	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
32	31	eqcomi 2746	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -((log ↾ (𝐴[,]𝐵))‘𝑥))
33	32	negfcncf 24868	. . . . . . 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 24843	. . . . . 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 588	. . . . 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 257	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
38		ioossre 13324	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
39		ltso 11214	. . . . . . . 8 ⊢ < Or ℝ
40		soss 5550	. . . . . . . 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 13350	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
44	43, 14	sstrid 3934	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴(,)𝐵) ⊆ ℝ⁺)
45	44	sselda 3922	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
46	45	rprecred 12961	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
47	46	renegcld 11565	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -(1 / 𝑥) ∈ ℝ)
48	47	fmpttd 7059	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)):(𝐴(,)𝐵)⟶ℝ)
49	48	frnd 6668	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ran (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) ⊆ ℝ)
50		soss 5550	. . . . . . . 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 5549	. . . . . . 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 11379	. . . . . . . . 9 ⊢ -(1 / 𝑥) ∈ V
55		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))
56	54, 55	fnmpti 6633	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)) Fn (𝐴(,)𝐵)
57		dffn4 6750	. . . . . . . 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 230	. . . . . . 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 3922	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → 𝑧 ∈ ℝ⁺)
61	60	adantrl 717	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → 𝑧 ∈ ℝ⁺)
62	61	rprecred 12961	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑧) ∈ ℝ)
63	44	sselda 3922	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
64	63	adantrr 718	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → 𝑦 ∈ ℝ⁺)
65	64	rprecred 12961	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (1 / 𝑦) ∈ ℝ)
66	62, 65	ltnegd 11716	. . . . . . . . 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 12968	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ (𝑦 ∈ (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵))) → (𝑦 < 𝑧 ↔ (1 / 𝑧) < (1 / 𝑦)))
68		oveq2 7366	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
69	68	negeqd 11375	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → -(1 / 𝑥) = -(1 / 𝑦))
70		negex 11379	. . . . . . . . . . . 12 ⊢ -(1 / 𝑦) ∈ V
71	69, 55, 70	fvmpt 6939	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑦) = -(1 / 𝑦))
72		oveq2 7366	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (1 / 𝑥) = (1 / 𝑧))
73	72	negeqd 11375	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → -(1 / 𝑥) = -(1 / 𝑧))
74		negex 11379	. . . . . . . . . . . 12 ⊢ -(1 / 𝑧) ∈ V
75	73, 55, 74	fvmpt 6939	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐴(,)𝐵) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥))‘𝑧) = -(1 / 𝑧))
76	71, 75	breqan12d 5102	. . . . . . . . . 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 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)))
78	66, 67, 77	3bitr4d 311	. . . . . . . 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 229	. . . . . . 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 3181	. . . . . 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 7274	. . . . . 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 839	. . . . 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 11119	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
84	83	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ℝ ∈ {ℝ, ℂ})
85		relogcl 26524	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
86	85	adantl 481	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
87	86	recnd 11161	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
88	87	negcld 11480	. . . . . . 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 7393	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ V)
91		relogf1o 26515	. . . . . . . . . . . . 13 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
92		f1of 6772	. . . . . . . . . . . . 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 6900	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
95		fvres 6851	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
96	95	mpteq2ia 5181	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
97	94, 96	eqtrdi 2788	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
98	97	oveq2d 7374	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
99		dvrelog 26586	. . . . . . . . 9 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
100	98, 99	eqtr3di 2787	. . . . . . . 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 25911	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ -(log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ -(1 / 𝑥)))
102		tgioo4 24748	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
103		eqid 2737	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
104		iccntr 24765	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
105	2, 4, 104	syl2anc 585	. . . . . . 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 25907	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ -(1 / 𝑥)))
107		isoeq1 7263	. . . . . 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 257	. . . 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 484	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0(,)1))
111		eqid 2737	. . . 4 ⊢ ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))
112	2, 4, 5, 37, 109, 110, 111	dvcvx 25966	. . 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 11085	. . . . . . . 8 ⊢ 1 ∈ ℂ
114		elioore 13292	. . . . . . . . . 10 ⊢ (𝑇 ∈ (0(,)1) → 𝑇 ∈ ℝ)
115	114	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℝ)
116	115	recnd 11161	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℂ)
117		nncan 11411	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
118	113, 116, 117	sylancr 588	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − (1 − 𝑇)) = 𝑇)
119	118	oveq1d 7373	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − (1 − 𝑇)) · 𝐴) = (𝑇 · 𝐴))
120	119	oveq1d 7373	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)))
121		ioossicc 13350	. . . . . . . 8 ⊢ (0(,)1) ⊆ (0[,]1)
122	121, 110	sselid 3920	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0[,]1))
123		iirev 24874	. . . . . . 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 13412	. . . . . 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 1376	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
127	120, 126	eqeltrrd 2838	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
128		fveq2 6832	. . . . . 6 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → (log‘𝑥) = (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
129	128	negeqd 11375	. . . . 5 ⊢ (𝑥 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) → -(log‘𝑥) = -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
130		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))
131		negex 11379	. . . . 5 ⊢ -(log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ V
132	129, 130, 131	fvmpt 6939	. . . 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 12951	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ^*)
135	3	rpxrd 12951	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ^*)
136	2, 4, 5	ltled 11282	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ≤ 𝐵)
137		lbicc2 13381	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138	134, 135, 136, 137	syl3anc 1374	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ (𝐴[,]𝐵))
139		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
140	139	negeqd 11375	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → -(log‘𝑥) = -(log‘𝐴))
141		negex 11379	. . . . . . . . 9 ⊢ -(log‘𝐴) ∈ V
142	140, 130, 141	fvmpt 6939	. . . . . . . 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 7374	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = (𝑇 · -(log‘𝐴)))
145	1	relogcld 26572	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℝ)
146	145	recnd 11161	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐴) ∈ ℂ)
147	116, 146	mulneg2d 11592	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · -(log‘𝐴)) = -(𝑇 · (log‘𝐴)))
148	144, 147	eqtrd 2772	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐴)) = -(𝑇 · (log‘𝐴)))
149		ubicc2 13382	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
150	134, 135, 136, 149	syl3anc 1374	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ (𝐴[,]𝐵))
151		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
152	151	negeqd 11375	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → -(log‘𝑥) = -(log‘𝐵))
153		negex 11379	. . . . . . . . 9 ⊢ -(log‘𝐵) ∈ V
154	152, 130, 153	fvmpt 6939	. . . . . . . 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 7374	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((𝑥 ∈ (𝐴[,]𝐵) ↦ -(log‘𝑥))‘𝐵)) = ((1 − 𝑇) · -(log‘𝐵)))
157		1re 11133	. . . . . . . . 9 ⊢ 1 ∈ ℝ
158		resubcl 11446	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
159	157, 115, 158	sylancr 588	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℝ)
160	159	recnd 11161	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ ℂ)
161	3	relogcld 26572	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℝ)
162	161	recnd 11161	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘𝐵) ∈ ℂ)
163	160, 162	mulneg2d 11592	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · -(log‘𝐵)) = -((1 − 𝑇) · (log‘𝐵)))
164	156, 163	eqtrd 2772	. . . . 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 7376	. . . 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 11163	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℝ)
167	166	recnd 11161	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · (log‘𝐴)) ∈ ℂ)
168	159, 161	remulcld 11163	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℝ)
169	168	recnd 11161	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · (log‘𝐵)) ∈ ℂ)
170	167, 169	negdid 11506	. . . 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 2775	. . 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 5117	. 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 11162	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) ∈ ℝ)
174	14, 127	sseldd 3923	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ ℝ⁺)
175	174	relogcld 26572	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) ∈ ℝ)
176	173, 175	ltnegd 11716	. 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 257	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ⊆ wss 3890 {cpr 4570 class class class wbr 5086 ↦ cmpt 5167 Po wpo 5528 Or wor 5529 ran crn 5623 ↾ cres 5624 Fn wfn 6485 ⟶wf 6486 –onto→wfo 6488 –1-1-onto→wf1o 6489 ‘cfv 6490 Isom wiso 6491 (class class class)co 7358 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 +∞cpnf 11164 ℝ^*cxr 11166 < clt 11167 ≤ cle 11168 − cmin 11365 -cneg 11366 / cdiv 11795 ℝ⁺crp 12906 (,)cioo 13262 [,]cicc 13265 TopOpenctopn 17342 topGenctg 17358 ℂ_fldccnfld 21311 intcnt 22960 –cn→ccncf 24821 D cdv 25808 logclog 26503

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-q 12863 df-rp 12907 df-xneg 13027 df-xadd 13028 df-xmul 13029 df-ioo 13266 df-ioc 13267 df-ico 13268 df-icc 13269 df-fz 13425 df-fzo 13572 df-fl 13713 df-mod 13791 df-seq 13926 df-exp 13986 df-fac 14198 df-bc 14227 df-hash 14255 df-shft 14991 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-limsup 15395 df-clim 15412 df-rlim 15413 df-sum 15611 df-ef 15991 df-sin 15993 df-cos 15994 df-pi 15996 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-starv 17193 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-unif 17201 df-hom 17202 df-cco 17203 df-rest 17343 df-topn 17344 df-0g 17362 df-gsum 17363 df-topgen 17364 df-pt 17365 df-prds 17368 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18710 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24263 df-ms 24264 df-tms 24265 df-cncf 24823 df-limc 25811 df-dv 25812 df-log 26505

This theorem is referenced by: amgmlem 26940 amgmwlem 50235

W3C validator