dvbdfbdioolem2 - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioolem2 44944

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 7086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
3	2	recnd 11247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
4	3	abscld 15388	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
5		dvbdfbdioolem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	5	rexrd 11269	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
7		dvbdfbdioolem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
8	7	rexrd 11269	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	5, 7	readdcld 11248	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
10	9	rehalfcld 12464	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
11		dvbdfbdioolem2.altb	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
12		avglt1 12455	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
13	5, 7, 12	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
14	11, 13	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < ((𝐴 + 𝐵) / 2))
15		avglt2 12456	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
16	5, 7, 15	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
17	11, 16	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) < 𝐵)
18	6, 8, 10, 14, 17	eliood 44510	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
19	1, 18	ffvelcdmd 7087	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
20	19	recnd 11247	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
21	20	abscld 15388	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
23	4, 22	resubcld 11647	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ∈ ℝ)
24		dvbdfbdioolem2.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ)
25	24	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐾 ∈ ℝ)
26	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
27	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
28	26, 27	resubcld 11647	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐵 − 𝐴) ∈ ℝ)
29	25, 28	remulcld 11249	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐾 · (𝐵 − 𝐴)) ∈ ℝ)
30	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
31	3, 30	subcld 11576	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
32	31	abscld 15388	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ∈ ℝ)
33	3, 30	abs2difd 15409	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))))
34		simpll 764	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝜑)
35	10	rexrd 11269	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ^*)
36	35	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ^*)
37	8	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝐵 ∈ ℝ^*)
38		elioore 13359	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
39	38	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
40	39	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ ℝ)
41		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) < 𝑥)
42	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
43	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
44		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
45		iooltub 44522	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 < 𝐵)
46	42, 43, 44, 45	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 < 𝐵)
47	46	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 < 𝐵)
48	36, 37, 40, 41, 47	eliood 44510	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵))
49	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐴 ∈ ℝ)
50	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐵 ∈ ℝ)
51	1	adantr 480	. . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
54	24	adantr 480	. . . . . . . . 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 6896	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
57	56	breq1d 5158	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾))
58	57	cbvralvw 3233	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
59	55, 58	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
60	59	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
61	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
62		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵))
63	49, 50, 51, 53, 54, 60, 61, 62	dvbdfbdioolem1 44943	. . . . . . . 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 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
65	34, 48, 64	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
66		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝐵) / 2) = 𝑥 → (𝐹‘((𝐴 + 𝐵) / 2)) = (𝐹‘𝑥))
67	66	eqcomd 2737	. . . . . . . . . . . . 13 ⊢ (((𝐴 + 𝐵) / 2) = 𝑥 → (𝐹‘𝑥) = (𝐹‘((𝐴 + 𝐵) / 2)))
68	67	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘𝑥) = (𝐹‘((𝐴 + 𝐵) / 2)))
69	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
70	68, 69	eqeltrd 2832	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘𝑥) ∈ ℂ)
71	70, 68	subeq0bd 11645	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) = 0)
72	71	abs00bd 15243	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) = 0)
73	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐾 ∈ ℝ)
74	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐵 ∈ ℝ)
75	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐴 ∈ ℝ)
76	74, 75	resubcld 11647	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐵 − 𝐴) ∈ ℝ)
77		0red 11222	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
78		ioossre 13390	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ ℝ
79		dvfre 25704	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
80	1, 78, 79	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
81	18, 52	eleqtrrd 2835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ dom (ℝ D 𝐹))
82	80, 81	ffvelcdmd 7087	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
83	82	recnd 11247	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
84	83	abscld 15388	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
85	83	absge0d 15396	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
86		2fveq3 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
87	86	breq1d 5158	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾))
88	87	rspccva 3611	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾)
90	77, 84, 24, 85, 89	letrd 11376	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐾)
91	90	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ 𝐾)
92	7, 5	resubcld 11647	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
93	5, 7	posdifd 11806	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
94	11, 93	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
95	77, 92, 94	ltled 11367	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴))
96	95	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐵 − 𝐴))
97	73, 76, 91, 96	mulge0d 11796	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐾 · (𝐵 − 𝐴)))
98	72, 97	eqbrtrd 5170	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
99	98	ad4ant14 749	. . . . . . 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 764	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)))
101	39	ad2antrr 723	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 ∈ ℝ)
102	10	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
103	39	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ ℝ)
104	10	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
105		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → ¬ ((𝐴 + 𝐵) / 2) < 𝑥)
106	103, 104, 105	nltled 11369	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ≤ ((𝐴 + 𝐵) / 2))
107	106	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 ≤ ((𝐴 + 𝐵) / 2))
108		neqne 2947	. . . . . . . . . 10 ⊢ (¬ ((𝐴 + 𝐵) / 2) = 𝑥 → ((𝐴 + 𝐵) / 2) ≠ 𝑥)
109	108	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐴 + 𝐵) / 2) ≠ 𝑥)
110	101, 102, 107, 109	leneltd 11373	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 < ((𝐴 + 𝐵) / 2))
111	3, 30	abssubd 15405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))))
112	111	adantr 480	. . . . . . . . 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 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐴 ∈ ℝ)
114	7	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐵 ∈ ℝ)
115	1	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
116	52	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
117	24	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐾 ∈ ℝ)
118	59	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
119	44	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 ∈ (𝐴(,)𝐵))
120	38	rexrd 11269	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ^*)
121	120	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 ∈ ℝ^*)
122	8	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐵 ∈ ℝ^*)
123	10	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
124		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 < ((𝐴 + 𝐵) / 2))
125	17	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) < 𝐵)
126	121, 122, 123, 124, 125	eliood 44510	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) ∈ (𝑥(,)𝐵))
127	113, 114, 115, 116, 117, 118, 119, 126	dvbdfbdioolem1 44943	. . . . . . . . . 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 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))) ≤ (𝐾 · (𝐵 − 𝐴)))
129	112, 128	eqbrtrd 5170	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
130	100, 110, 129	syl2anc 583	. . . . . . 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 810	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
132	65, 131	pm2.61dan 810	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
133	23, 32, 29, 33, 132	letrd 11376	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
134	23, 29, 22, 133	leadd1dd 11833	. . 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 11247	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℂ)
136	22	recnd 11247	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
137	135, 136	npcand 11580	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘(𝐹‘𝑥)))
138	137	eqcomd 2737	. . 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 11247	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
141	24	recnd 11247	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ)
142	7	recnd 11247	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
143	5	recnd 11247	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
144	142, 143	subcld 11576	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
145	141, 144	mulcld 11239	. . . . . 6 ⊢ (𝜑 → (𝐾 · (𝐵 − 𝐴)) ∈ ℂ)
146	140, 145	addcomd 11421	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴))) = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
147	139, 146	eqtrid 2783	. . . 4 ⊢ (𝜑 → 𝑀 = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
148	147	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑀 = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
149	134, 138, 148	3brtr4d 5180	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ≤ 𝑀)
150	149	ralrimiva 3145	1 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ⊆ wss 3948 class class class wbr 5148 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 ℝcr 11113 0cc0 11114 + caddc 11117 · cmul 11119 ℝ^*cxr 11252 < clt 11253 ≤ cle 11254 − cmin 11449 / cdiv 11876 2c2 12272 (,)cioo 13329 abscabs 15186 D cdv 25613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-cmp 23112 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617

This theorem is referenced by: dvbdfbdioo 44945

W3C validator