dvbdfbdioolem2 - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioolem2 46464

Description: A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
dvbdfbdioolem2.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvbdfbdioolem2.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvbdfbdioolem2.altb	⊢ (𝜑 → 𝐴 < 𝐵)
dvbdfbdioolem2.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
dvbdfbdioolem2.dmdv	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvbdfbdioolem2.k	⊢ (𝜑 → 𝐾 ∈ ℝ)
dvbdfbdioolem2.dvbd	⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
dvbdfbdioolem2.m	⊢ 𝑀 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴)))

**Assertion**
Ref	Expression
dvbdfbdioolem2	⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑀)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝑥,𝐾 𝜑,𝑥 Allowed substitution hint: 𝑀(𝑥)

Proof of Theorem dvbdfbdioolem2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvbdfbdioolem2.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
2	1	ffvelcdmda 7060	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
3	2	recnd 11204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
4	3	abscld 15457	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
5		dvbdfbdioolem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	5	rexrd 11226	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
7		dvbdfbdioolem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
8	7	rexrd 11226	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	5, 7	readdcld 11205	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
10	9	rehalfcld 12462	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
11		dvbdfbdioolem2.altb	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
12		avglt1 12453	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
13	5, 7, 12	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
14	11, 13	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < ((𝐴 + 𝐵) / 2))
15		avglt2 12454	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
16	5, 7, 15	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
17	11, 16	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) < 𝐵)
18	6, 8, 10, 14, 17	eliood 46035	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
19	1, 18	ffvelcdmd 7061	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
20	19	recnd 11204	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
21	20	abscld 15457	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
22	21	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
23	4, 22	resubcld 11609	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ∈ ℝ)
24		dvbdfbdioolem2.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ)
25	24	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐾 ∈ ℝ)
26	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
27	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
28	26, 27	resubcld 11609	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐵 − 𝐴) ∈ ℝ)
29	25, 28	remulcld 11206	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐾 · (𝐵 − 𝐴)) ∈ ℝ)
30	20	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
31	3, 30	subcld 11536	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
32	31	abscld 15457	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ∈ ℝ)
33	3, 30	abs2difd 15478	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))))
34		simpll 776	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝜑)
35	10	rexrd 11226	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ^*)
36	35	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ^*)
37	8	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝐵 ∈ ℝ^*)
38		elioore 13373	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
39	38	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
40	39	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ ℝ)
41		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) < 𝑥)
42	6	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
43	8	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
44		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
45		iooltub 46047	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 < 𝐵)
46	42, 43, 44, 45	syl3anc 1389	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 < 𝐵)
47	46	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 < 𝐵)
48	36, 37, 40, 41, 47	eliood 46035	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵))
49	5	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐴 ∈ ℝ)
50	7	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐵 ∈ ℝ)
51	1	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
52		dvbdfbdioolem2.dmdv	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
53	52	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
54	24	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐾 ∈ ℝ)
55		dvbdfbdioolem2.dvbd	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
56		2fveq3 6867	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
57	56	breq1d 5107	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾))
58	57	cbvralvw 3239	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
59	55, 58	sylib 220	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
60	59	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
61	18	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
62		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵))
63	49, 50, 51, 53, 54, 60, 61, 62	dvbdfbdioolem1 46463	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → ((abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝑥 − ((𝐴 + 𝐵) / 2))) ∧ (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴))))
64	63	simprd 499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
65	34, 48, 64	syl2anc 593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
66		fveq2 6862	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝐵) / 2) = 𝑥 → (𝐹‘((𝐴 + 𝐵) / 2)) = (𝐹‘𝑥))
67	66	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ (((𝐴 + 𝐵) / 2) = 𝑥 → (𝐹‘𝑥) = (𝐹‘((𝐴 + 𝐵) / 2)))
68	67	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘𝑥) = (𝐹‘((𝐴 + 𝐵) / 2)))
69	20	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
70	68, 69	eqeltrd 2861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘𝑥) ∈ ℂ)
71	70, 68	subeq0bd 11607	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) = 0)
72	71	abs00bd 15309	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) = 0)
73	24	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐾 ∈ ℝ)
74	7	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐵 ∈ ℝ)
75	5	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐴 ∈ ℝ)
76	74, 75	resubcld 11609	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐵 − 𝐴) ∈ ℝ)
77		0red 11178	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
78		ioossre 13405	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ ℝ
79		dvfre 26001	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
80	1, 78, 79	sylancl 595	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
81	18, 52	eleqtrrd 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ dom (ℝ D 𝐹))
82	80, 81	ffvelcdmd 7061	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
83	82	recnd 11204	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
84	83	abscld 15457	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
85	83	absge0d 15465	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
86		2fveq3 6867	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
87	86	breq1d 5107	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾))
88	87	rspccva 3579	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ∧ ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾)
89	55, 18, 88	syl2anc 593	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾)
90	77, 84, 24, 85, 89	letrd 11334	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐾)
91	90	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ 𝐾)
92	7, 5	resubcld 11609	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
93	5, 7	posdifd 11768	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
94	11, 93	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
95	77, 92, 94	ltled 11325	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴))
96	95	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐵 − 𝐴))
97	73, 76, 91, 96	mulge0d 11758	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐾 · (𝐵 − 𝐴)))
98	72, 97	eqbrtrd 5119	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
99	98	ad4ant14 762	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
100		simpll 776	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)))
101	39	ad2antrr 736	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 ∈ ℝ)
102	10	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
103	39	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ ℝ)
104	10	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
105		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → ¬ ((𝐴 + 𝐵) / 2) < 𝑥)
106	103, 104, 105	nltled 11327	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ≤ ((𝐴 + 𝐵) / 2))
107	106	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 ≤ ((𝐴 + 𝐵) / 2))
108		neqne 2964	. . . . . . . . . 10 ⊢ (¬ ((𝐴 + 𝐵) / 2) = 𝑥 → ((𝐴 + 𝐵) / 2) ≠ 𝑥)
109	108	adantl 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐴 + 𝐵) / 2) ≠ 𝑥)
110	101, 102, 107, 109	leneltd 11331	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 < ((𝐴 + 𝐵) / 2))
111	3, 30	abssubd 15474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))))
112	111	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))))
113	5	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐴 ∈ ℝ)
114	7	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐵 ∈ ℝ)
115	1	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
116	52	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
117	24	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐾 ∈ ℝ)
118	59	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
119	44	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 ∈ (𝐴(,)𝐵))
120	38	rexrd 11226	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ^*)
121	120	ad2antlr 737	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 ∈ ℝ^*)
122	8	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐵 ∈ ℝ^*)
123	10	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
124		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 < ((𝐴 + 𝐵) / 2))
125	17	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) < 𝐵)
126	121, 122, 123, 124, 125	eliood 46035	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) ∈ (𝑥(,)𝐵))
127	113, 114, 115, 116, 117, 118, 119, 126	dvbdfbdioolem1 46463	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))) ≤ (𝐾 · (((𝐴 + 𝐵) / 2) − 𝑥)) ∧ (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))) ≤ (𝐾 · (𝐵 − 𝐴))))
128	127	simprd 499	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))) ≤ (𝐾 · (𝐵 − 𝐴)))
129	112, 128	eqbrtrd 5119	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
130	100, 110, 129	syl2anc 593	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
131	99, 130	pm2.61dan 822	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
132	65, 131	pm2.61dan 822	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
133	23, 32, 29, 33, 132	letrd 11334	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
134	23, 29, 22, 133	leadd1dd 11795	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
135	4	recnd 11204	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℂ)
136	22	recnd 11204	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
137	135, 136	npcand 11540	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘(𝐹‘𝑥)))
138	137	eqcomd 2767	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) = (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
139		dvbdfbdioolem2.m	. . . . 5 ⊢ 𝑀 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴)))
140	21	recnd 11204	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
141	24	recnd 11204	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ)
142	7	recnd 11204	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
143	5	recnd 11204	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
144	142, 143	subcld 11536	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
145	141, 144	mulcld 11196	. . . . . 6 ⊢ (𝜑 → (𝐾 · (𝐵 − 𝐴)) ∈ ℂ)
146	140, 145	addcomd 11379	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴))) = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
147	139, 146	eqtrid 2808	. . . 4 ⊢ (𝜑 → 𝑀 = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
148	147	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑀 = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
149	134, 138, 148	3brtr4d 5129	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ≤ 𝑀)
150	149	ralrimiva 3153	1 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ⊆ wss 3902 class class class wbr 5097 dom cdm 5643 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ℂcc 11065 ℝcr 11066 0cc0 11067 + caddc 11070 · cmul 11072 ℝ^*cxr 11209 < clt 11210 ≤ cle 11211 − cmin 11408 / cdiv 11838 2c2 12266 (,)cioo 13343 abscabs 15252 D cdv 25913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-ioo 13347 df-ico 13349 df-icc 13350 df-fz 13507 df-fzo 13654 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-rest 17442 df-topn 17443 df-0g 17461 df-gsum 17462 df-topgen 17463 df-pt 17464 df-prds 17467 df-xrs 17523 df-qtop 17528 df-imas 17529 df-xps 17531 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19101 df-cntz 19348 df-cmn 19813 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-fbas 21409 df-fg 21410 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-nei 23146 df-lp 23184 df-perf 23185 df-cn 23275 df-cnp 23276 df-haus 23363 df-cmp 23435 df-tx 23610 df-hmeo 23803 df-fil 23894 df-fm 23986 df-flim 23987 df-flf 23988 df-xms 24368 df-ms 24369 df-tms 24370 df-cncf 24928 df-limc 25916 df-dv 25917

This theorem is referenced by: dvbdfbdioo 46465

W3C validator