dvbdfbdioolem1 - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioolem1 46374

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 13351	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℝ
2		dvbdfbdioolem1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
3	1, 2	sselid 3920	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4		ioossre 13351	. . . 4 ⊢ (𝐶(,)𝐵) ⊆ ℝ
5		dvbdfbdioolem1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)𝐵))
6	4, 5	sselid 3920	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ)
7	3	rexrd 11186	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
8		dvbdfbdioolem1.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	rexrd 11186	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		ioogtlb 45943	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ (𝐶(,)𝐵)) → 𝐶 < 𝐷)
11	7, 9, 5, 10	syl3anc 1374	. . 3 ⊢ (𝜑 → 𝐶 < 𝐷)
12		dvbdfbdioolem1.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13	12	rexrd 11186	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
14		ioogtlb 45943	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
15	13, 9, 2, 14	syl3anc 1374	. . . . 5 ⊢ (𝜑 → 𝐴 < 𝐶)
16		iooltub 45958	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ (𝐶(,)𝐵)) → 𝐷 < 𝐵)
17	7, 9, 5, 16	syl3anc 1374	. . . . 5 ⊢ (𝜑 → 𝐷 < 𝐵)
18		iccssioo 13359	. . . . 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 11086	. . . . . . 7 ⊢ ℝ ⊆ ℂ
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
23	20, 22	fssd 6679	. . . . . . 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 25898	. . . . . . 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 1376	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
28		cncfcdm 24875	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
29	22, 27, 28	syl2anc 585	. . . . 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 24874	. . . 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 3933	. . . . . . 7 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
34		eqid 2737	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
35		tgioo4 24780	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
36	34, 35	dvres 25888	. . . . . . 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 24797	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
39	3, 6, 38	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
40	39	reseq2d 5938	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
41	37, 40	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
42	41	dmeqd 5854	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
43	12, 3, 15	ltled 11285	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
44	6, 8, 17	ltled 11285	. . . . . . 7 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
45		ioossioo 13385	. . . . . . 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 3960	. . . . 5 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹))
48		ssdmres 5972	. . . . 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 2772	. . 3 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝐶(,)𝐷))
51	3, 6, 11, 32, 50	mvth 25969	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐶(,)𝐷)((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)))
52	41	fveq1d 6836	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥))
53		fvres 6853	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐶(,)𝐷) → (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥) = ((ℝ D 𝐹)‘𝑥))
54	52, 53	sylan9eq 2792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((ℝ D 𝐹)‘𝑥))
55	54	eqcomd 2743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥))
56	55	3adant3 1133	. . . . . 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 1139	. . . . . 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 11186	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
59	3, 6, 11	ltled 11285	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
60		ubicc2 13409	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐷 ∈ (𝐶[,]𝐷))
61	7, 58, 59, 60	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (𝐶[,]𝐷))
62		fvres 6853	. . . . . . . . . 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 13408	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐶 ∈ (𝐶[,]𝐷))
65	7, 58, 59, 64	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (𝐶[,]𝐷))
66		fvres 6853	. . . . . . . . . 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 7378	. . . . . . . 8 ⊢ (𝜑 → (((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) = ((𝐹‘𝐷) − (𝐹‘𝐶)))
69	68	oveq1d 7375	. . . . . . 7 ⊢ (𝜑 → ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
70	69	3ad2ant1 1134	. . . . . 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 2776	. . . . 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 1139	. . . . . . . . . . 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 2743	. . . . . . . . . 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 3923	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (𝐴(,)𝐵))
75	20, 74	ffvelcdmd 7031	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℝ)
76	20, 2	ffvelcdmd 7031	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
77	75, 76	resubcld 11569	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℝ)
78	77	recnd 11164	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℂ)
79	78	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℂ)
80		dvfre 25928	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
81	20, 24, 80	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
82	25	feq2d 6646	. . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝑥 ∈ (𝐴(,)𝐵))
86	84, 85	ffvelcdmd 7031	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
87	86	recnd 11164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
88	87	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
89	6, 3	resubcld 11569	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
90	89	recnd 11164	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℂ)
91	90	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (𝐷 − 𝐶) ∈ ℂ)
92	3, 6	posdifd 11728	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 < 𝐷 ↔ 0 < (𝐷 − 𝐶)))
93	11, 92	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐷 − 𝐶))
94	93	gt0ne0d 11705	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 − 𝐶) ≠ 0)
95	94	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (𝐷 − 𝐶) ≠ 0)
96	79, 88, 91, 95	divmul3d 11956	. . . . . . . . . 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 6838	. . . . . . . 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 15410	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))))
101	100	3adant3 1133	. . . . . . . 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 2772	. . . . . . 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 15387	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) = (𝐷 − 𝐶))
104	103	oveq2d 7376	. . . . . . . 8 ⊢ (𝜑 → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)))
105	104	3ad2ant1 1134	. . . . . . 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 2772	. . . . . 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 15392	. . . . . . . 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 11138	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
112	111, 89, 93	ltled 11285	. . . . . . . . 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 3227	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
117	115, 85, 116	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
118	107, 109, 110, 113, 117	lemul1ad 12086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)) ≤ (𝐾 · (𝐷 − 𝐶)))
119	118	3adant3 1133	. . . . . 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 5108	. . . . 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 1412	. . . 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 15392	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ∈ ℝ)
123	8, 12	resubcld 11569	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
124	123	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐵 − 𝐴) ∈ ℝ)
125	87	absge0d 15400	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑥)))
126	99	absge0d 15400	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘(𝐷 − 𝐶)))
127	6, 12, 8, 3, 44, 43	le2subd 11761	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 − 𝐶) ≤ (𝐵 − 𝐴))
128	103, 127	eqbrtrd 5108	. . . . . . . . 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 12089	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
131	130	3adant3 1133	. . . . . 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 5108	. . . . 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 1412	. . . 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 3143	. 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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ⊆ wss 3890 class class class wbr 5086 dom cdm 5624 ran crn 5625 ↾ cres 5626 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 · cmul 11034 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 (,)cioo 13289 [,]cicc 13292 abscabs 15187 TopOpenctopn 17375 topGenctg 17391 ℂ_fldccnfld 21344 intcnt 22992 –cn→ccncf 24853 D cdv 25840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844

This theorem is referenced by: dvbdfbdioolem2 46375 ioodvbdlimc1lem1 46377 ioodvbdlimc1lem2 46378 ioodvbdlimc2lem 46380

W3C validator