dvbdfbdioolem1 - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioolem1 45883

Description: Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
dvbdfbdioolem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvbdfbdioolem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvbdfbdioolem1.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
dvbdfbdioolem1.dmdv	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvbdfbdioolem1.k	⊢ (𝜑 → 𝐾 ∈ ℝ)
dvbdfbdioolem1.dvbd	⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
dvbdfbdioolem1.c	⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
dvbdfbdioolem1.d	⊢ (𝜑 → 𝐷 ∈ (𝐶(,)𝐵))

**Assertion**
Ref	Expression
dvbdfbdioolem1	⊢ (𝜑 → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐹 𝑥,𝐾 𝜑,𝑥

**Proof of Theorem dvbdfbdioolem1**
Step	Hyp	Ref	Expression
1		ioossre 13444	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℝ
2		dvbdfbdioolem1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
3	1, 2	sselid 3992	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4		ioossre 13444	. . . 4 ⊢ (𝐶(,)𝐵) ⊆ ℝ
5		dvbdfbdioolem1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)𝐵))
6	4, 5	sselid 3992	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ)
7	3	rexrd 11308	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
8		dvbdfbdioolem1.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	rexrd 11308	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		ioogtlb 45447	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ (𝐶(,)𝐵)) → 𝐶 < 𝐷)
11	7, 9, 5, 10	syl3anc 1370	. . 3 ⊢ (𝜑 → 𝐶 < 𝐷)
12		dvbdfbdioolem1.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13	12	rexrd 11308	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
14		ioogtlb 45447	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
15	13, 9, 2, 14	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝐴 < 𝐶)
16		iooltub 45462	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ (𝐶(,)𝐵)) → 𝐷 < 𝐵)
17	7, 9, 5, 16	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝐷 < 𝐵)
18		iccssioo 13452	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
19	13, 9, 15, 17, 18	syl22anc 839	. . . 4 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
20		dvbdfbdioolem1.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
21		ax-resscn 11209	. . . . . . 7 ⊢ ℝ ⊆ ℂ
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
23	20, 22	fssd 6753	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
24	1	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
25		dvbdfbdioolem1.dmdv	. . . . . . 7 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
26		dvcn 25971	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D 𝐹) = (𝐴(,)𝐵)) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
27	22, 23, 24, 25, 26	syl31anc 1372	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
28		cncfcdm 24937	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
29	22, 27, 28	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
30	20, 29	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
31		rescncf 24936	. . . 4 ⊢ ((𝐶[,]𝐷) ⊆ (𝐴(,)𝐵) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐹 ↾ (𝐶[,]𝐷)) ∈ ((𝐶[,]𝐷)–cn→ℝ)))
32	19, 30, 31	sylc 65	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) ∈ ((𝐶[,]𝐷)–cn→ℝ))
33	19, 24	sstrd 4005	. . . . . . 7 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
34		eqid 2734	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
35	34	tgioo2 24838	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
36	34, 35	dvres 25960	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ) ∧ ((𝐴(,)𝐵) ⊆ ℝ ∧ (𝐶[,]𝐷) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))))
37	22, 23, 24, 33, 36	syl22anc 839	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))))
38		iccntr 24856	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
39	3, 6, 38	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
40	39	reseq2d 5999	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
41	37, 40	eqtrd 2774	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
42	41	dmeqd 5918	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
43	12, 3, 15	ltled 11406	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
44	6, 8, 17	ltled 11406	. . . . . . 7 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
45		ioossioo 13477	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
46	13, 9, 43, 44, 45	syl22anc 839	. . . . . 6 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
47	46, 25	sseqtrrd 4036	. . . . 5 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹))
48		ssdmres 6032	. . . . 5 ⊢ ((𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝐶(,)𝐷))
49	47, 48	sylib 218	. . . 4 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝐶(,)𝐷))
50	42, 49	eqtrd 2774	. . 3 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝐶(,)𝐷))
51	3, 6, 11, 32, 50	mvth 26045	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐶(,)𝐷)((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)))
52	41	fveq1d 6908	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥))
53		fvres 6925	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐶(,)𝐷) → (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥) = ((ℝ D 𝐹)‘𝑥))
54	52, 53	sylan9eq 2794	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((ℝ D 𝐹)‘𝑥))
55	54	eqcomd 2740	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥))
56	55	3adant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥))
57		simp3 1137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)))
58	6	rexrd 11308	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
59	3, 6, 11	ltled 11406	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
60		ubicc2 13501	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐷 ∈ (𝐶[,]𝐷))
61	7, 58, 59, 60	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (𝐶[,]𝐷))
62		fvres 6925	. . . . . . . . . 10 ⊢ (𝐷 ∈ (𝐶[,]𝐷) → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) = (𝐹‘𝐷))
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) = (𝐹‘𝐷))
64		lbicc2 13500	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐶 ∈ (𝐶[,]𝐷))
65	7, 58, 59, 64	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (𝐶[,]𝐷))
66		fvres 6925	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝐶[,]𝐷) → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶) = (𝐹‘𝐶))
67	65, 66	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶) = (𝐹‘𝐶))
68	63, 67	oveq12d 7448	. . . . . . . 8 ⊢ (𝜑 → (((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) = ((𝐹‘𝐷) − (𝐹‘𝐶)))
69	68	oveq1d 7445	. . . . . . 7 ⊢ (𝜑 → ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
70	69	3ad2ant1 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
71	56, 57, 70	3eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
72		simp3 1137	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
73	72	eqcomd 2740	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)) = ((ℝ D 𝐹)‘𝑥))
74	19, 61	sseldd 3995	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (𝐴(,)𝐵))
75	20, 74	ffvelcdmd 7104	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℝ)
76	20, 2	ffvelcdmd 7104	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
77	75, 76	resubcld 11688	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℝ)
78	77	recnd 11286	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℂ)
79	78	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℂ)
80		dvfre 26003	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
81	20, 24, 80	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
82	25	feq2d 6722	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
83	81, 82	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
84	83	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
85	46	sselda 3994	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝑥 ∈ (𝐴(,)𝐵))
86	84, 85	ffvelcdmd 7104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
87	86	recnd 11286	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
88	87	3adant3 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
89	6, 3	resubcld 11688	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
90	89	recnd 11286	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℂ)
91	90	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (𝐷 − 𝐶) ∈ ℂ)
92	3, 6	posdifd 11847	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 < 𝐷 ↔ 0 < (𝐷 − 𝐶)))
93	11, 92	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐷 − 𝐶))
94	93	gt0ne0d 11824	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 − 𝐶) ≠ 0)
95	94	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (𝐷 − 𝐶) ≠ 0)
96	79, 88, 91, 95	divmul3d 12074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)) = ((ℝ D 𝐹)‘𝑥) ↔ ((𝐹‘𝐷) − (𝐹‘𝐶)) = (((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))))
97	73, 96	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((𝐹‘𝐷) − (𝐹‘𝐶)) = (((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶)))
98	97	fveq2d 6910	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))))
99	90	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐷 − 𝐶) ∈ ℂ)
100	87, 99	absmuld 15489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))))
101	100	3adant3 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))))
102	98, 101	eqtrd 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))))
103	3, 6, 59	abssubge0d 15466	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) = (𝐷 − 𝐶))
104	103	oveq2d 7446	. . . . . . . 8 ⊢ (𝜑 → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)))
105	104	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)))
106	102, 105	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)))
107	87	abscld 15471	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ∈ ℝ)
108		dvbdfbdioolem1.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ)
109	108	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝐾 ∈ ℝ)
110	89	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐷 − 𝐶) ∈ ℝ)
111		0red 11261	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
112	111, 89, 93	ltled 11406	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐷 − 𝐶))
113	112	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (𝐷 − 𝐶))
114		dvbdfbdioolem1.dvbd	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
115	114	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
116		rspa 3245	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
117	115, 85, 116	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
118	107, 109, 110, 113, 117	lemul1ad 12204	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)) ≤ (𝐾 · (𝐷 − 𝐶)))
119	118	3adant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)) ≤ (𝐾 · (𝐷 − 𝐶)))
120	106, 119	eqbrtrd 5169	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)))
121	71, 120	syld3an3 1408	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)))
122	99	abscld 15471	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ∈ ℝ)
123	8, 12	resubcld 11688	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
124	123	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐵 − 𝐴) ∈ ℝ)
125	87	absge0d 15479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑥)))
126	99	absge0d 15479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘(𝐷 − 𝐶)))
127	6, 12, 8, 3, 44, 43	le2subd 11880	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 − 𝐶) ≤ (𝐵 − 𝐴))
128	103, 127	eqbrtrd 5169	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) ≤ (𝐵 − 𝐴))
129	128	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ≤ (𝐵 − 𝐴))
130	107, 109, 122, 124, 125, 126, 117, 129	lemul12ad 12207	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
131	130	3adant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
132	102, 131	eqbrtrd 5169	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
133	71, 132	syld3an3 1408	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
134	121, 133	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))))
135	134	rexlimdv3a 3156	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝐶(,)𝐷)((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))))
136	51, 135	mpd 15	1 ⊢ (𝜑 → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 class class class wbr 5147 dom cdm 5688 ran crn 5689 ↾ cres 5690 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 · cmul 11157 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 − cmin 11489 / cdiv 11917 (,)cioo 13383 [,]cicc 13386 abscabs 15269 TopOpenctopn 17467 topGenctg 17483 ℂ_fldccnfld 21381 intcnt 23040 –cn→ccncf 24915 D cdv 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 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 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 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916

This theorem is referenced by: dvbdfbdioolem2 45884 ioodvbdlimc1lem1 45886 ioodvbdlimc1lem2 45887 ioodvbdlimc2lem 45889

W3C validator