dvbdfbdioolem2 - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioolem2 45944

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 7104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
3	2	recnd 11289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
4	3	abscld 15475	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
5		dvbdfbdioolem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	5	rexrd 11311	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
7		dvbdfbdioolem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
8	7	rexrd 11311	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	5, 7	readdcld 11290	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
10	9	rehalfcld 12513	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
11		dvbdfbdioolem2.altb	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
12		avglt1 12504	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
13	5, 7, 12	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
14	11, 13	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < ((𝐴 + 𝐵) / 2))
15		avglt2 12505	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
16	5, 7, 15	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
17	11, 16	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) < 𝐵)
18	6, 8, 10, 14, 17	eliood 45511	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
19	1, 18	ffvelcdmd 7105	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
20	19	recnd 11289	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
21	20	abscld 15475	. . . . . 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 11691	. . . 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 11691	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐵 − 𝐴) ∈ ℝ)
29	25, 28	remulcld 11291	. . . 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 11620	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
32	31	abscld 15475	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ∈ ℝ)
33	3, 30	abs2difd 15496	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))))
34		simpll 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝜑)
35	10	rexrd 11311	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ^*)
36	35	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ^*)
37	8	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝐵 ∈ ℝ^*)
38		elioore 13417	. . . . . . . . . 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 45523	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 < 𝐵)
46	42, 43, 44, 45	syl3anc 1373	. . . . . . . . 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 45511	. . . . . . 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 6911	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
57	56	breq1d 5153	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾))
58	57	cbvralvw 3237	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
59	55, 58	sylib 218	. . . . . . . . . 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 45943	. . . . . . . 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 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
66		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝐵) / 2) = 𝑥 → (𝐹‘((𝐴 + 𝐵) / 2)) = (𝐹‘𝑥))
67	66	eqcomd 2743	. . . . . . . . . . . . 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 2841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘𝑥) ∈ ℂ)
71	70, 68	subeq0bd 11689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) = 0)
72	71	abs00bd 15330	. . . . . . . . 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 11691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐵 − 𝐴) ∈ ℝ)
77		0red 11264	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
78		ioossre 13448	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ ℝ
79		dvfre 25989	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
80	1, 78, 79	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
81	18, 52	eleqtrrd 2844	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ dom (ℝ D 𝐹))
82	80, 81	ffvelcdmd 7105	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
83	82	recnd 11289	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
84	83	abscld 15475	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
85	83	absge0d 15483	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
86		2fveq3 6911	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
87	86	breq1d 5153	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾))
88	87	rspccva 3621	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾)
90	77, 84, 24, 85, 89	letrd 11418	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐾)
91	90	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ 𝐾)
92	7, 5	resubcld 11691	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
93	5, 7	posdifd 11850	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
94	11, 93	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
95	77, 92, 94	ltled 11409	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴))
96	95	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐵 − 𝐴))
97	73, 76, 91, 96	mulge0d 11840	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐾 · (𝐵 − 𝐴)))
98	72, 97	eqbrtrd 5165	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
99	98	ad4ant14 752	. . . . . . 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 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)))
101	39	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 ∈ ℝ)
102	10	ad3antrrr 730	. . . . . . . . 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 726	. . . . . . . . . . 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 11411	. . . . . . . . . 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 2948	. . . . . . . . . 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 11415	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 < ((𝐴 + 𝐵) / 2))
111	3, 30	abssubd 15492	. . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐴 ∈ ℝ)
114	7	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐵 ∈ ℝ)
115	1	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
116	52	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
117	24	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐾 ∈ ℝ)
118	59	ad2antrr 726	. . . . . . . . . . 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 11311	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ^*)
121	120	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 ∈ ℝ^*)
122	8	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐵 ∈ ℝ^*)
123	10	ad2antrr 726	. . . . . . . . . . . 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 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) < 𝐵)
126	121, 122, 123, 124, 125	eliood 45511	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) ∈ (𝑥(,)𝐵))
127	113, 114, 115, 116, 117, 118, 119, 126	dvbdfbdioolem1 45943	. . . . . . . . . 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 5165	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
130	100, 110, 129	syl2anc 584	. . . . . . 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 813	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
132	65, 131	pm2.61dan 813	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
133	23, 32, 29, 33, 132	letrd 11418	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
134	23, 29, 22, 133	leadd1dd 11877	. . 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 11289	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℂ)
136	22	recnd 11289	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
137	135, 136	npcand 11624	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘(𝐹‘𝑥)))
138	137	eqcomd 2743	. . 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 11289	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
141	24	recnd 11289	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ)
142	7	recnd 11289	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
143	5	recnd 11289	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
144	142, 143	subcld 11620	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
145	141, 144	mulcld 11281	. . . . . 6 ⊢ (𝜑 → (𝐾 · (𝐵 − 𝐴)) ∈ ℂ)
146	140, 145	addcomd 11463	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴))) = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
147	139, 146	eqtrid 2789	. . . 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 5175	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ≤ 𝑀)
150	149	ralrimiva 3146	1 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 + caddc 11158 · cmul 11160 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 (,)cioo 13387 abscabs 15273 D cdv 25898

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902

This theorem is referenced by: dvbdfbdioo 45945

W3C validator