efcvx - Metamath Proof Explorer

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Theorem efcvx 26416

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 1192	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ)
2		simpl2 1193	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ)
3		simpl3 1194	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 < 𝐵)
4		reeff1o 26414	. . . . . . 7 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
5		f1of 6823	. . . . . . 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 13021	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
8		fss 6727	. . . . . 6 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ)
9	6, 7, 8	mp2an 692	. . . . 5 ⊢ (exp ↾ ℝ):ℝ⟶ℝ
10		iccssre 13451	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
11	1, 2, 10	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℝ)
12		fssres2 6751	. . . . 5 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ) → (exp ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
13	9, 11, 12	sylancr 587	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (exp ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
14		ax-resscn 11191	. . . . 5 ⊢ ℝ ⊆ ℂ
15	11, 14	sstrdi 3976	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℂ)
16		efcn 26410	. . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ)
17		rescncf 24846	. . . . . 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 24847	. . . . 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 587	. . . 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 257	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (exp ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
22		reefiso 26415	. . . . . 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 13429	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ
25	24	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴(,)𝐵) ⊆ ℝ)
26		eqidd 2737	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ ℝ) “ (𝐴(,)𝐵)) = ((exp ↾ ℝ) “ (𝐴(,)𝐵)))
27		isores3 7333	. . . . 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 1373	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵))))
29		ssid 3986	. . . . . . 7 ⊢ ℝ ⊆ ℝ
30		fss 6727	. . . . . . . . 9 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → (exp ↾ ℝ):ℝ⟶ℂ)
31	9, 14, 30	mp2an 692	. . . . . . . 8 ⊢ (exp ↾ ℝ):ℝ⟶ℂ
32		eqid 2736	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
33		tgioo4 24749	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
34	32, 33	dvres 25869	. . . . . . . 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 702	. . . . . . 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 587	. . . . . 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 6000	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ ℝ) ↾ (𝐴[,]𝐵)) = (exp ↾ (𝐴[,]𝐵)))
38	37	oveq2d 7426	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D ((exp ↾ ℝ) ↾ (𝐴[,]𝐵))) = (ℝ D (exp ↾ (𝐴[,]𝐵))))
39		reelprrecn 11226	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
40		eff 16102	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
41		ssid 3986	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
42		dvef 25941	. . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp
43	42	dmeqi 5889	. . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp
44	40	fdmi 6722	. . . . . . . . . . . 12 ⊢ dom exp = ℂ
45	43, 44	eqtri 2759	. . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ
46	14, 45	sseqtrri 4013	. . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp)
47		dvres3 25871	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ)
49	42	reseq1i 5967	. . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ)
50	48, 49	eqtri 2759	. . . . . . . 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 24766	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
53	1, 2, 52	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
54	51, 53	reseq12d 5972	. . . . . 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 2779	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (exp ↾ (𝐴[,]𝐵))) = ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)))
56		isoeq1 7315	. . . . 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 257	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (exp ↾ (𝐴[,]𝐵))) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵))))
59		simpr 484	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0(,)1))
60		eqid 2736	. . 3 ⊢ ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))
61	1, 2, 3, 21, 58, 59, 60	dvcvx 25982	. 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 11192	. . . . . . 7 ⊢ 1 ∈ ℂ
63		ioossre 13429	. . . . . . . . 9 ⊢ (0(,)1) ⊆ ℝ
64	63, 59	sselid 3961	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℝ)
65	64	recnd 11268	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℂ)
66		nncan 11517	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
67	62, 65, 66	sylancr 587	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (1 − (1 − 𝑇)) = 𝑇)
68	67	oveq1d 7425	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − (1 − 𝑇)) · 𝐴) = (𝑇 · 𝐴))
69	68	oveq1d 7425	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)))
70		ioossicc 13455	. . . . . . 7 ⊢ (0(,)1) ⊆ (0[,]1)
71	70, 59	sselid 3961	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0[,]1))
72		iirev 24879	. . . . . 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 13517	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ (1 − 𝑇) ∈ (0[,]1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
75	73, 74	syldan 591	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
76	69, 75	eqeltrrd 2836	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
77	76	fvresd 6901	. 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 11290	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ^*)
79	2	rexrd 11290	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ^*)
80	1, 2, 3	ltled 11388	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ≤ 𝐵)
81		lbicc2 13486	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
82	78, 79, 80, 81	syl3anc 1373	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ (𝐴[,]𝐵))
83	82	fvresd 6901	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵))‘𝐴) = (exp‘𝐴))
84	83	oveq2d 7426	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((exp ↾ (𝐴[,]𝐵))‘𝐴)) = (𝑇 · (exp‘𝐴)))
85		ubicc2 13487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
86	78, 79, 80, 85	syl3anc 1373	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ (𝐴[,]𝐵))
87	86	fvresd 6901	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵))‘𝐵) = (exp‘𝐵))
88	87	oveq2d 7426	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((exp ↾ (𝐴[,]𝐵))‘𝐵)) = ((1 − 𝑇) · (exp‘𝐵)))
89	84, 88	oveq12d 7428	. 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 5155	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⊆ wss 3931 {cpr 4608 class class class wbr 5124 dom cdm 5659 ran crn 5660 ↾ cres 5661 “ cima 5662 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 Isom wiso 6537 (class class class)co 7410 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 − cmin 11471 ℝ⁺crp 13013 (,)cioo 13367 [,]cicc 13370 expce 16082 TopOpenctopn 17440 topGenctg 17456 ℂ_fldccnfld 21320 intcnt 22960 –cn→ccncf 24825 D cdv 25821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 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 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825

This theorem is referenced by: (None)

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