dvbdfbdioolem2 - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioolem2 45900

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 7038	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
3	2	recnd 11178	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
4	3	abscld 15381	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
5		dvbdfbdioolem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	5	rexrd 11200	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
7		dvbdfbdioolem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
8	7	rexrd 11200	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	5, 7	readdcld 11179	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
10	9	rehalfcld 12405	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
11		dvbdfbdioolem2.altb	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
12		avglt1 12396	. . . . . . . . . . . 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 12397	. . . . . . . . . . . 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 45469	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
19	1, 18	ffvelcdmd 7039	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
20	19	recnd 11178	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
21	20	abscld 15381	. . . . . 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 11582	. . . 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 11582	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐵 − 𝐴) ∈ ℝ)
29	25, 28	remulcld 11180	. . . 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 11509	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
32	31	abscld 15381	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ∈ ℝ)
33	3, 30	abs2difd 15402	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))))
34		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝜑)
35	10	rexrd 11200	. . . . . . . . 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 13312	. . . . . . . . . 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 45481	. . . . . . . . . 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 45469	. . . . . . 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 6845	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
57	56	breq1d 5112	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾))
58	57	cbvralvw 3213	. . . . . . . . . . 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 45899	. . . . . . . 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 6840	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝐵) / 2) = 𝑥 → (𝐹‘((𝐴 + 𝐵) / 2)) = (𝐹‘𝑥))
67	66	eqcomd 2735	. . . . . . . . . . . . 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 2828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘𝑥) ∈ ℂ)
71	70, 68	subeq0bd 11580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) = 0)
72	71	abs00bd 15233	. . . . . . . . 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 11582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐵 − 𝐴) ∈ ℝ)
77		0red 11153	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
78		ioossre 13344	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ ℝ
79		dvfre 25831	. . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ dom (ℝ D 𝐹))
82	80, 81	ffvelcdmd 7039	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
83	82	recnd 11178	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
84	83	abscld 15381	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
85	83	absge0d 15389	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
86		2fveq3 6845	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
87	86	breq1d 5112	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾))
88	87	rspccva 3584	. . . . . . . . . . . . 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 11307	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐾)
91	90	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ 𝐾)
92	7, 5	resubcld 11582	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
93	5, 7	posdifd 11741	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
94	11, 93	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
95	77, 92, 94	ltled 11298	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴))
96	95	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐵 − 𝐴))
97	73, 76, 91, 96	mulge0d 11731	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐾 · (𝐵 − 𝐴)))
98	72, 97	eqbrtrd 5124	. . . . . . . 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 766	. . . . . . . 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 11300	. . . . . . . . . 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 2933	. . . . . . . . . 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 11304	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 < ((𝐴 + 𝐵) / 2))
111	3, 30	abssubd 15398	. . . . . . . . . 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 11200	. . . . . . . . . . . . 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 45469	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) ∈ (𝑥(,)𝐵))
127	113, 114, 115, 116, 117, 118, 119, 126	dvbdfbdioolem1 45899	. . . . . . . . . 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 5124	. . . . . . . 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 812	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
132	65, 131	pm2.61dan 812	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
133	23, 32, 29, 33, 132	letrd 11307	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
134	23, 29, 22, 133	leadd1dd 11768	. . 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 11178	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℂ)
136	22	recnd 11178	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
137	135, 136	npcand 11513	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘(𝐹‘𝑥)))
138	137	eqcomd 2735	. . 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 11178	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
141	24	recnd 11178	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ)
142	7	recnd 11178	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
143	5	recnd 11178	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
144	142, 143	subcld 11509	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
145	141, 144	mulcld 11170	. . . . . 6 ⊢ (𝜑 → (𝐾 · (𝐵 − 𝐴)) ∈ ℂ)
146	140, 145	addcomd 11352	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴))) = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
147	139, 146	eqtrid 2776	. . . 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 5134	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ≤ 𝑀)
150	149	ralrimiva 3125	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 2109 ≠ wne 2925 ∀wral 3044 ⊆ wss 3911 class class class wbr 5102 dom cdm 5631 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 + caddc 11047 · cmul 11049 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 2c2 12217 (,)cioo 13282 abscabs 15176 D cdv 25740

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744

This theorem is referenced by: dvbdfbdioo 45901

W3C validator