efcvx - Metamath Proof Explorer

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Theorem efcvx 26494

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 1191	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ)
2		simpl2 1192	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ)
3		simpl3 1193	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 < 𝐵)
4		reeff1o 26492	. . . . . . 7 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
5		f1of 6847	. . . . . . 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 13043	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
8		fss 6751	. . . . . 6 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ)
9	6, 7, 8	mp2an 692	. . . . 5 ⊢ (exp ↾ ℝ):ℝ⟶ℝ
10		iccssre 13470	. . . . . 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 6775	. . . . 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 11213	. . . . 5 ⊢ ℝ ⊆ ℂ
15	11, 14	sstrdi 3995	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝐴[,]𝐵) ⊆ ℂ)
16		efcn 26488	. . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ)
17		rescncf 24924	. . . . . 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 24925	. . . . 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 26493	. . . . . 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 13449	. . . . . 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 7356	. . . . 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 1372	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)) Isom < , < ((𝐴(,)𝐵), ((exp ↾ ℝ) “ (𝐴(,)𝐵))))
29		ssid 4005	. . . . . . 7 ⊢ ℝ ⊆ ℝ
30		fss 6751	. . . . . . . . 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 24827	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
34	32, 33	dvres 25947	. . . . . . . 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 6025	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ ℝ) ↾ (𝐴[,]𝐵)) = (exp ↾ (𝐴[,]𝐵)))
38	37	oveq2d 7448	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D ((exp ↾ ℝ) ↾ (𝐴[,]𝐵))) = (ℝ D (exp ↾ (𝐴[,]𝐵))))
39		reelprrecn 11248	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
40		eff 16118	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
41		ssid 4005	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
42		dvef 26019	. . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp
43	42	dmeqi 5914	. . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp
44	40	fdmi 6746	. . . . . . . . . . . 12 ⊢ dom exp = ℂ
45	43, 44	eqtri 2764	. . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ
46	14, 45	sseqtrri 4032	. . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp)
47		dvres3 25949	. . . . . . . . . 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 5992	. . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ)
50	48, 49	eqtri 2764	. . . . . . . 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 24844	. . . . . . . 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 5997	. . . . . 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 2784	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (ℝ D (exp ↾ (𝐴[,]𝐵))) = ((exp ↾ ℝ) ↾ (𝐴(,)𝐵)))
56		isoeq1 7338	. . . . 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 26060	. 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 11214	. . . . . . 7 ⊢ 1 ∈ ℂ
63		ioossre 13449	. . . . . . . . 9 ⊢ (0(,)1) ⊆ ℝ
64	63, 59	sselid 3980	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℝ)
65	64	recnd 11290	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ ℂ)
66		nncan 11539	. . . . . . 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 7447	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − (1 − 𝑇)) · 𝐴) = (𝑇 · 𝐴))
69	68	oveq1d 7447	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (((1 − (1 − 𝑇)) · 𝐴) + ((1 − 𝑇) · 𝐵)) = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)))
70		ioossicc 13474	. . . . . . 7 ⊢ (0(,)1) ⊆ (0[,]1)
71	70, 59	sselid 3980	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝑇 ∈ (0[,]1))
72		iirev 24957	. . . . . 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 13536	. . . . 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 2841	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ (𝐴[,]𝐵))
77	76	fvresd 6925	. 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 11312	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ ℝ^*)
79	2	rexrd 11312	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ ℝ^*)
80	1, 2, 3	ltled 11410	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ≤ 𝐵)
81		lbicc2 13505	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
82	78, 79, 80, 81	syl3anc 1372	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐴 ∈ (𝐴[,]𝐵))
83	82	fvresd 6925	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵))‘𝐴) = (exp‘𝐴))
84	83	oveq2d 7448	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → (𝑇 · ((exp ↾ (𝐴[,]𝐵))‘𝐴)) = (𝑇 · (exp‘𝐴)))
85		ubicc2 13506	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
86	78, 79, 80, 85	syl3anc 1372	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → 𝐵 ∈ (𝐴[,]𝐵))
87	86	fvresd 6925	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((exp ↾ (𝐴[,]𝐵))‘𝐵) = (exp‘𝐵))
88	87	oveq2d 7448	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((1 − 𝑇) · ((exp ↾ (𝐴[,]𝐵))‘𝐵)) = ((1 − 𝑇) · (exp‘𝐵)))
89	84, 88	oveq12d 7450	. 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 5173	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 1539 ∈ wcel 2107 ⊆ wss 3950 {cpr 4627 class class class wbr 5142 dom cdm 5684 ran crn 5685 ↾ cres 5686 “ cima 5687 ⟶wf 6556 –1-1-onto→wf1o 6559 ‘cfv 6560 Isom wiso 6561 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 ℝ^*cxr 11295 < clt 11296 ≤ cle 11297 − cmin 11493 ℝ⁺crp 13035 (,)cioo 13388 [,]cicc 13391 expce 16098 TopOpenctopn 17467 topGenctg 17483 ℂ_fldccnfld 21365 intcnt 23026 –cn→ccncf 24903 D cdv 25899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903

This theorem is referenced by: (None)

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