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Theorem c1liplem1 26309

Description: Lemma for c1lip1 26310. (Contributed by Stefan O'Rear, 15-Nov-2014.)

**Hypotheses**
Ref	Expression
c1liplem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
c1liplem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
c1liplem1.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
c1liplem1.f	⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
c1liplem1.dv	⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
c1liplem1.cn	⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
c1liplem1.k	⊢ 𝐾 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )

**Assertion**
Ref	Expression
c1liplem1	⊢ (𝜑 → (𝐾 ∈ ℝ ∧ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥))))))

Distinct variable groups: 𝜑,𝑥,𝑦 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 𝑥,𝐹,𝑦 Allowed substitution hints: 𝐾(𝑥, 𝑦)

Proof of Theorem c1liplem1 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		c1liplem1.k	. . 3 ⊢ 𝐾 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )
2		imassrn 6196	. . . . . 6 ⊢ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ran abs
3		absf 15498	. . . . . . 7 ⊢ abs:ℂ⟶ℝ
4		frn 6715	. . . . . . 7 ⊢ (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
5	3, 4	ax-mp 5	. . . . . 6 ⊢ ran abs ⊆ ℝ
6	2, 5	sstri 3940	. . . . 5 ⊢ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ
7	6	a1i 11	. . . 4 ⊢ (𝜑 → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ)
8		dvf 26220	. . . . . . . 8 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
9		ffun 6710	. . . . . . . 8 ⊢ ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ → Fun (ℝ D 𝐹))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ Fun (ℝ D 𝐹)
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → Fun (ℝ D 𝐹))
12		c1liplem1.dv	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
13		cncff 25207	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
14		fdm 6717	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ → dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
15	12, 13, 14	3syl 19	. . . . . . 7 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
16		ssdmres 6004	. . . . . . 7 ⊢ ((𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
17	15, 16	sylibr 237	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
18		c1liplem1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19	18	rexrd 11352	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
20		c1liplem1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
21	20	rexrd 11352	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
22		c1liplem1.le	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
23		lbicc2 13588	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
24	19, 21, 22, 23	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
25		funfvima2 7235	. . . . . . 7 ⊢ ((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) → (𝐴 ∈ (𝐴[,]𝐵) → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
26	25	imp 412	. . . . . 6 ⊢ (((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) ∧ 𝐴 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
27	11, 17, 24, 26	syl21anc 851	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
28		ffun 6710	. . . . . . 7 ⊢ (abs:ℂ⟶ℝ → Fun abs)
29	3, 28	ax-mp 5	. . . . . 6 ⊢ Fun abs
30		imassrn 6196	. . . . . . . 8 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ ran (ℝ D 𝐹)
31		frn 6715	. . . . . . . . 9 ⊢ ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ → ran (ℝ D 𝐹) ⊆ ℂ)
32	8, 31	ax-mp 5	. . . . . . . 8 ⊢ ran (ℝ D 𝐹) ⊆ ℂ
33	30, 32	sstri 3940	. . . . . . 7 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ ℂ
34	3	fdmi 6719	. . . . . . 7 ⊢ dom abs = ℂ
35	33, 34	sseqtrri 3980	. . . . . 6 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ dom abs
36		funfvima2 7235	. . . . . 6 ⊢ ((Fun abs ∧ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ dom abs) → (((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
37	29, 35, 36	mp2an 705	. . . . 5 ⊢ (((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
38		ne0i 4287	. . . . 5 ⊢ ((abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
39	27, 37, 38	3syl 19	. . . 4 ⊢ (𝜑 → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
40		ax-resscn 11250	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
41		ssid 3953	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
42		cncfss 25213	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
43	40, 41, 42	mp2an 705	. . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
44	43, 12	sselid 3929	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
45		cniccbdd 25775	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎)
46	18, 20, 44, 45	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎)
47		fvelima 6948	. . . . . . . . . 10 ⊢ ((Fun abs ∧ 𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))) → ∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏)
48	29, 47	mpan 703	. . . . . . . . 9 ⊢ (𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏)
49		fvres 6902	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (𝐴[,]𝐵) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏) = ((ℝ D 𝐹)‘𝑏))
50	49	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏) = ((ℝ D 𝐹)‘𝑏))
51	50	fveq2d 6887	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) = (abs‘((ℝ D 𝐹)‘𝑏)))
52		2fveq3 6888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) = (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)))
53	52	breq1d 5113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → ((abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ↔ (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) ≤ 𝑎))
54	53	rspccva 3576	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) ≤ 𝑎)
55	51, 54	eqbrtrrd 5129	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎)
56	55	adantll 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎)
57		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑏)) = (abs‘𝑦))
58	57	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ (((ℝ D 𝐹)‘𝑏) = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎 ↔ (abs‘𝑦) ≤ 𝑎))
59	56, 58	syl5ibcom 248	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘𝑦) ≤ 𝑎))
60	59	rexlimdva 3164	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘𝑦) ≤ 𝑎))
61		fvelima 6948	. . . . . . . . . . . . 13 ⊢ ((Fun (ℝ D 𝐹) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦)
62	10, 61	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → ∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦)
63	60, 62	impel 515	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs‘𝑦) ≤ 𝑎)
64		breq1 5106	. . . . . . . . . . 11 ⊢ ((abs‘𝑦) = 𝑏 → ((abs‘𝑦) ≤ 𝑎 ↔ 𝑏 ≤ 𝑎))
65	63, 64	syl5ibcom 248	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ((abs‘𝑦) = 𝑏 → 𝑏 ≤ 𝑎))
66	65	rexlimdva 3164	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏 → 𝑏 ≤ 𝑎))
67	48, 66	syl5 35	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → 𝑏 ≤ 𝑎))
68	67	ralrimiv 3154	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎)
69	68	ex 418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 → ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎))
70	69	reximdva 3176	. . . . 5 ⊢ (𝜑 → (∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎))
71	46, 70	mpd 16	. . . 4 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎)
72	7, 39, 71	suprcld 12273	. . 3 ⊢ (𝜑 → sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ)
73	1, 72	eqeltrid 2865	. 2 ⊢ (𝜑 → 𝐾 ∈ ℝ)
74		simplrr 790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝐴[,]𝐵))
75	74	fvresd 6903	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦))
76		c1liplem1.cn	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
77		cncff 25207	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
78	76, 77	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
79	78	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
80	79, 74	ffvelcdmd 7083	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
81	80	recnd 11330	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℂ)
82	75, 81	eqeltrrd 2862	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑦) ∈ ℂ)
83		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝐴[,]𝐵))
84	83	fvresd 6903	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
85	79, 83	ffvelcdmd 7083	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
86	85	recnd 11330	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℂ)
87	84, 86	eqeltrrd 2862	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) ∈ ℂ)
88	82, 87	subcld 11662	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑦) − (𝐹‘𝑥)) ∈ ℂ)
89		iccssre 13553	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
90	18, 20, 89	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
91	90	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ ℝ)
92	91, 74	sseldd 3932	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
93	91, 83	sseldd 3932	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
94	92, 93	resubcld 11737	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ)
95	94	recnd 11330	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℂ)
96		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
97		difrp 13153	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℝ⁺))
98	93, 92, 97	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℝ⁺))
99	96, 98	mpbid 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ⁺)
100	99	rpne0d 13162	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ≠ 0)
101	88, 95, 100	absdivd 15618	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) = ((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (abs‘(𝑦 − 𝑥))))
102	6	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ)
103	39	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
104	71	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎)
105	29	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → Fun abs)
106	88, 95, 100	divcld 12086	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ℂ)
107	106, 34	eleqtrrdi 2872	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ dom abs)
108	93	rexrd 11352	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ^*)
109	92	rexrd 11352	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ^*)
110	93, 92, 96	ltled 11451	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦)
111		ubicc2 13589	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ (𝑥[,]𝑦))
112	108, 109, 110, 111	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝑥[,]𝑦))
113	112	fvresd 6903	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) = (𝐹‘𝑦))
114		lbicc2 13588	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑥 ≤ 𝑦) → 𝑥 ∈ (𝑥[,]𝑦))
115	108, 109, 110, 114	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝑥[,]𝑦))
116	115	fvresd 6903	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥) = (𝐹‘𝑥))
117	113, 116	oveq12d 7436	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) = ((𝐹‘𝑦) − (𝐹‘𝑥)))
118	117	oveq1d 7433	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) = (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)))
119		iccss2 13541	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥[,]𝑦) ⊆ (𝐴[,]𝐵))
120	119	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥[,]𝑦) ⊆ (𝐴[,]𝐵))
121	120	resabs1d 5999	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) = (𝐹 ↾ (𝑥[,]𝑦)))
122	76	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
123		rescncf 25211	. . . . . . . . . . . . . . 15 ⊢ ((𝑥[,]𝑦) ⊆ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ)))
124	120, 122, 123	sylc 66	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ))
125	121, 124	eqeltrrd 2862	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ))
126	40	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ℝ ⊆ ℂ)
127		c1liplem1.f	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
128	127	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹 ∈ (ℂ ↑_pm ℝ))
129		cnex 11274	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ ∈ V
130		reex 11284	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ∈ V
131	129, 130	elpm2 8895	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
132	131	simplbi 502	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
133	128, 132	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹:dom 𝐹⟶ℂ)
134	131	simprbi 503	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → dom 𝐹 ⊆ ℝ)
135	128, 134	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom 𝐹 ⊆ ℝ)
136		iccssre 13553	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥[,]𝑦) ⊆ ℝ)
137	93, 92, 136	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥[,]𝑦) ⊆ ℝ)
138		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
139		tgioo4 25117	. . . . . . . . . . . . . . . . . 18 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
140	138, 139	dvres 26224	. . . . . . . . . . . . . . . . 17 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:dom 𝐹⟶ℂ) ∧ (dom 𝐹 ⊆ ℝ ∧ (𝑥[,]𝑦) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))))
141	126, 133, 135, 137, 140	syl22anc 852	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))))
142		iccntr 25134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
143	93, 92, 142	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
144	143	reseq2d 5970	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
145	141, 144	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
146	145	dmeqd 5887	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
147		ioossicc 13557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(,)𝑦) ⊆ (𝑥[,]𝑦)
148	147, 120	sstrid 3942	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥(,)𝑦) ⊆ (𝐴[,]𝐵))
149	17	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
150	148, 149	sstrd 3941	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥(,)𝑦) ⊆ dom (ℝ D 𝐹))
151		ssdmres 6004	. . . . . . . . . . . . . . 15 ⊢ ((𝑥(,)𝑦) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)) = (𝑥(,)𝑦))
152	150, 151	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)) = (𝑥(,)𝑦))
153	146, 152	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = (𝑥(,)𝑦))
154	93, 92, 96, 125, 153	mvth 26305	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ∃𝑎 ∈ (𝑥(,)𝑦)((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)))
155	145	fveq1d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎))
156	155	adantrr 730	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎))
157		fvres 6902	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑥(,)𝑦) → (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
158	157	ad2antll 742	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
159	156, 158	eqtrd 2796	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
160	10	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → Fun (ℝ D 𝐹))
161	17	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
162	148	sseld 3930	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑎 ∈ (𝑥(,)𝑦) → 𝑎 ∈ (𝐴[,]𝐵)))
163	162	impr 460	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → 𝑎 ∈ (𝐴[,]𝐵))
164		funfvima2 7235	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) → (𝑎 ∈ (𝐴[,]𝐵) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
165	164	imp 412	. . . . . . . . . . . . . . . . 17 ⊢ (((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) ∧ 𝑎 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
166	160, 161, 163, 165	syl21anc 851	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
167	159, 166	eqeltrd 2861	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
168		eleq1 2849	. . . . . . . . . . . . . . 15 ⊢ (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ↔ ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
169	167, 168	syl5ibcom 248	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
170	169	expr 462	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑎 ∈ (𝑥(,)𝑦) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
171	170	rexlimdv 3162	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∃𝑎 ∈ (𝑥(,)𝑦)((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
172	154, 171	mpd 16	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
173	118, 172	eqeltrrd 2862	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
174		funfvima 7234	. . . . . . . . . . 11 ⊢ ((Fun abs ∧ (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ dom abs) → ((((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
175	174	imp 412	. . . . . . . . . 10 ⊢ (((Fun abs ∧ (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ dom abs) ∧ (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
176	105, 107, 173, 175	syl21anc 851	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
177	102, 103, 104, 176	suprubd 12272	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ≤ sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ))
178	177, 1	breqtrrdi 5147	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ≤ 𝐾)
179	101, 178	eqbrtrrd 5129	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (abs‘(𝑦 − 𝑥))) ≤ 𝐾)
180	88	abscld 15599	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ∈ ℝ)
181	73	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℝ)
182	95, 100	absrpcld 15611	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(𝑦 − 𝑥)) ∈ ℝ⁺)
183	180, 181, 182	ledivmuld 13210	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (abs‘(𝑦 − 𝑥))) ≤ 𝐾 ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ ((abs‘(𝑦 − 𝑥)) · 𝐾)))
184	179, 183	mpbid 235	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ ((abs‘(𝑦 − 𝑥)) · 𝐾))
185	182	rpcnd 13159	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(𝑦 − 𝑥)) ∈ ℂ)
186	181	recnd 11330	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℂ)
187	185, 186	mulcomd 11323	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((abs‘(𝑦 − 𝑥)) · 𝐾) = (𝐾 · (abs‘(𝑦 − 𝑥))))
188	184, 187	breqtrd 5131	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥))))
189	188	ex 418	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥)))))
190	189	ralrimivva 3206	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥)))))
191	73, 190	jca 521	1 ⊢ (𝜑 → (𝐾 ∈ ℝ ∧ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 ⊆ wss 3899 ∅c0 4279 class class class wbr 5103 dom cdm 5651 ran crn 5652 ↾ cres 5653 “ cima 5654 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ↑_pm cpm 8841 supcsup 9425 ℂcc 11191 ℝcr 11192 · cmul 11198 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 − cmin 11534 / cdiv 11966 ℝ⁺crp 13113 (,)cioo 13469 [,]cicc 13472 abscabs 15394 TopOpenctopn 17585 topGenctg 17601 ℂ_fldccnfld 21671 intcnt 23328 –cn→ccncf 25190 D cdv 26176

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cn 23538 df-cnp 23539 df-haus 23626 df-cmp 23698 df-tx 23874 df-hmeo 24067 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-limc 26179 df-dv 26180

This theorem is used by: c1lip1 26310

W3C validator