efcvx - Metamath Proof Explorer

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Theorem efcvx 26439

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

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

**Proof of Theorem efcvx**
Step	Hyp	Ref	Expression
1		simpl1 1198	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ)
2		simpl2 1199	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ)
3		simpl3 1200	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 < 𝐵)
4		reeff1o 26437	. . . . . . 7 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
5		f1of 6774	. . . . . . 7 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ (exp ↾ ℝ):ℝ⟶ℝ⁺
7		rpssre 12948	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
8		fss 6678	. . . . . 6 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ)
9	6, 7, 8	mp2an 698	. . . . 5 ⊢ (exp ↾ ℝ):ℝ⟶ℝ
10		iccssre 13380	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
11	1, 2, 10	syl2anc 590	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℝ)
12		fssres2 6702	. . . . 5 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ) → (exp ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
13	9, 11, 12	sylancr 593	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (exp ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
14		ax-resscn 11093	. . . . 5 ⊢ ℝ ⊆ ℂ
15	11, 14	sstrdi 3934	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℂ)
16		efcn 26433	. . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ)
17		rescncf 24889	. . . . . 6 ⊢ ((𝐴[,]𝐵) ⊆ ℂ → (exp ∈ (ℂ–cn→ℂ) → (exp ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
18	15, 16, 17	mpisyl 21	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (exp ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
19		cncfcdm 24890	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ (exp ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ((exp ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (exp ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ))
20	14, 18, 19	sylancr 593	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (exp ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ))
21	13, 20	mpbird 258	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (exp ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
22		reefiso 26438	. . . . . 6 ⊢ (exp ↾ ℝ) Isom < , < (ℝ, ℝ⁺)
23	22	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (exp ↾ ℝ) Isom < , < (ℝ, ℝ⁺))
24		ioossre 13358	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ
25	24	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴(,)𝐵) ⊆ ℝ)
26		eqidd 2741	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ ℝ) “ (𝐴(,)𝐵)) = ((exp ↾ ℝ) “ (𝐴(,)𝐵)))
27		isores3 7286	. . . . 5 ⊢ (((exp ↾ ℝ) Isom < , < (ℝ, ℝ⁺) ∧ (𝐴(,)𝐵) ⊆ ℝ ∧ ((exp ↾ ℝ) “ (𝐴(,)𝐵)) = ((exp ↾ ℝ) “ (𝐴(,)𝐵))) → ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵))))
28	23, 25, 26, 27	syl3anc 1379	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵))))
29		ssid 3944	. . . . . . 7 ⊢ ℝ ⊆ ℝ
30		fss 6678	. . . . . . . . 9 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → (exp ↾ ℝ):ℝ⟶ℂ)
31	9, 14, 30	mp2an 698	. . . . . . . 8 ⊢ (exp ↾ ℝ):ℝ⟶ℂ
32		eqid 2740	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
33		tgioo4 24795	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
34	32, 33	dvres 25903	. . . . . . . 8 ⊢ (((ℝ ⊆ ℂ ∧ (exp ↾ ℝ):ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ)) → (ℝ D ((exp ↾ ℝ) ↾ (𝐴[,]𝐵))) = ((ℝ D (exp ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
35	14, 31, 34	mpanl12 708	. . . . . . 7 ⊢ ((ℝ ⊆ ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ) → (ℝ D ((exp ↾ ℝ) ↾ (𝐴[,]𝐵))) = ((ℝ D (exp ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
36	29, 11, 35	sylancr 593	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D ((exp ↾ ℝ) ↾ (𝐴[,]𝐵))) = ((ℝ D (exp ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
37	11	resabs1d 5967	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ ℝ) ↾ (𝐴[,]𝐵)) = (exp ↾ (𝐴[,]𝐵)))
38	37	oveq2d 7379	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D ((exp ↾ ℝ) ↾ (𝐴[,]𝐵))) = (ℝ D (exp ↾ (𝐴[,]𝐵))))
39		reelprrecn 11128	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
40		eff 16044	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
41		ssid 3944	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
42		dvef 25972	. . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp
43	42	dmeqi 5853	. . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp
44	40	fdmi 6673	. . . . . . . . . . . 12 ⊢ dom exp = ℂ
45	43, 44	eqtri 2763	. . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ
46	14, 45	sseqtrri 3971	. . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp)
47		dvres3 25905	. . . . . . . . . 10 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ exp:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D exp))) → (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ))
48	39, 40, 41, 46, 47	mp4an 699	. . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ)
49	42	reseq1i 5934	. . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ)
50	48, 49	eqtri 2763	. . . . . . . 8 ⊢ (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ)
51	50	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ))
52		iccntr 24812	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
53	1, 2, 52	syl2anc 590	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
54	51, 53	reseq12d 5939	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((ℝ D (exp ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)))
55	36, 38, 54	3eqtr3d 2783	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (exp ↾ (𝐴[,]𝐵))) = ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)))
56		isoeq1 7268	. . . . 5 ⊢ ((ℝ D (exp ↾ (𝐴[,]𝐵))) = ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)) → ((ℝ D (exp ↾ (𝐴[,]𝐵))) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵))) ↔ ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵)))))
57	55, 56	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((ℝ D (exp ↾ (𝐴[,]𝐵))) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵))) ↔ ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵)))))
58	28, 57	mpbird 258	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (exp ↾ (𝐴[,]𝐵))) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵))))
59		simpr 485	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0(,)1))
60		eqid 2740	. . 3 ⊢ ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))
61	1, 2, 3, 21, 58, 59, 60	dvcvx 26012	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵))‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) < ((𝑇 · ((exp ↾ (𝐴[,]𝐵))‘𝐴)) + ((1 − 𝑇) · ((exp ↾ (𝐴[,]𝐵))‘𝐵))))
62		ax-1cn 11094	. . . . . . 7 ⊢ 1 ∈ ℂ
63		ioossre 13358	. . . . . . . . 9 ⊢ (0(,)1) ⊆ ℝ
64	63, 59	sselid 3920	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℝ)
65	64	recnd 11171	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℂ)
66		nncan 11421	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
67	62, 65, 66	sylancr 593	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − (1 − 𝑇)) = 𝑇)
68	67	oveq1d 7378	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − (1 − 𝑇)) · 𝐴) = (𝑇 · 𝐴))
69	68	oveq1d 7378	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)))
70		ioossicc 13384	. . . . . . 7 ⊢ (0(,)1) ⊆ (0[,]1)
71	70, 59	sselid 3920	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0[,]1))
72		iirev 24921	. . . . . 6 ⊢ (𝑇 ∈ (0[,]1) → (1 − 𝑇) ∈ (0[,]1))
73	71, 72	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − 𝑇) ∈ (0[,]1))
74		lincmb01cmp 13446	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ (1 − 𝑇) ∈ (0[,]1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
75	73, 74	syldan 597	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
76	69, 75	eqeltrrd 2841	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
77	76	fvresd 6854	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵))‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) = (exp‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
78	1	rexrd 11193	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ^*)
79	2	rexrd 11193	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ^*)
80	1, 2, 3	ltled 11292	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ≤ 𝐵)
81		lbicc2 13415	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
82	78, 79, 80, 81	syl3anc 1379	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ (𝐴[,]𝐵))
83	82	fvresd 6854	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵))‘𝐴) = (exp‘𝐴))
84	83	oveq2d 7379	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((exp ↾ (𝐴[,]𝐵))‘𝐴)) = (𝑇 · (exp‘𝐴)))
85		ubicc2 13416	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
86	78, 79, 80, 85	syl3anc 1379	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ (𝐴[,]𝐵))
87	86	fvresd 6854	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵))‘𝐵) = (exp‘𝐵))
88	87	oveq2d 7379	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((exp ↾ (𝐴[,]𝐵))‘𝐵)) = ((1 − 𝑇) · (exp‘𝐵)))
89	84, 88	oveq12d 7381	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · ((exp ↾ (𝐴[,]𝐵))‘𝐴)) + ((1 − 𝑇) · ((exp ↾ (𝐴[,]𝐵))‘𝐵))) = ((𝑇 · (exp‘𝐴)) + ((1 − 𝑇) · (exp‘𝐵))))
90	61, 77, 89	3brtr3d 5110	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (exp‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))) < ((𝑇 · (exp‘𝐴)) + ((1 − 𝑇) · (exp‘𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ⊆ wss 3890 {cpr 4564 class class class wbr 5079 dom cdm 5625 ran crn 5626 ↾ cres 5627 “ cima 5628 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 Isom wiso 6493 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 − cmin 11375 ℝ⁺crp 12940 (,)cioo 13296 [,]cicc 13299 expce 16024 TopOpenctopn 17382 topGenctg 17398 ℂ_fldccnfld 21354 intcnt 23007 –cn→ccncf 24868 D cdv 25855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-cmp 23377 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859

This theorem is referenced by: (None)

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