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Theorem c1liplem1 24595

Description: Lemma for c1lip1 24596. (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 5942	. . . . . 6 ⊢ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ran abs
3		absf 14699	. . . . . . 7 ⊢ abs:ℂ⟶ℝ
4		frn 6522	. . . . . . 7 ⊢ (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
5	3, 4	ax-mp 5	. . . . . 6 ⊢ ran abs ⊆ ℝ
6	2, 5	sstri 3978	. . . . 5 ⊢ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ
7	6	a1i 11	. . . 4 ⊢ (𝜑 → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ)
8		dvf 24507	. . . . . . . 8 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
9		ffun 6519	. . . . . . . 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 23503	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
14		fdm 6524	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ → dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
15	12, 13, 14	3syl 18	. . . . . . 7 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
16		ssdmres 5878	. . . . . . 7 ⊢ ((𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
17	15, 16	sylibr 236	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
18		c1liplem1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19	18	rexrd 10693	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
20		c1liplem1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
21	20	rexrd 10693	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
22		c1liplem1.le	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
23		lbicc2 12855	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
24	19, 21, 22, 23	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
25		funfvima2 6995	. . . . . . 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 409	. . . . . 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 835	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
28		ffun 6519	. . . . . . 7 ⊢ (abs:ℂ⟶ℝ → Fun abs)
29	3, 28	ax-mp 5	. . . . . 6 ⊢ Fun abs
30		imassrn 5942	. . . . . . . 8 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ ran (ℝ D 𝐹)
31		frn 6522	. . . . . . . . 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 3978	. . . . . . 7 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ ℂ
34	3	fdmi 6526	. . . . . . 7 ⊢ dom abs = ℂ
35	33, 34	sseqtrri 4006	. . . . . 6 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ dom abs
36		funfvima2 6995	. . . . . 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 690	. . . . 5 ⊢ (((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
38		ne0i 4302	. . . . 5 ⊢ ((abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
39	27, 37, 38	3syl 18	. . . 4 ⊢ (𝜑 → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
40		ax-resscn 10596	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
41		ssid 3991	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
42		cncfss 23509	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
43	40, 41, 42	mp2an 690	. . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
44	43, 12	sseldi 3967	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
45		cniccbdd 24064	. . . . . 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 1367	. . . . 5 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎)
47		fvelima 6733	. . . . . . . . . 10 ⊢ ((Fun abs ∧ 𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))) → ∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏)
48	29, 47	mpan 688	. . . . . . . . 9 ⊢ (𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏)
49		fvres 6691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (𝐴[,]𝐵) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏) = ((ℝ D 𝐹)‘𝑏))
50	49	adantl 484	. . . . . . . . . . . . . . . . 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 6676	. . . . . . . . . . . . . . . 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 6677	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) = (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)))
53	52	breq1d 5078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → ((abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ↔ (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) ≤ 𝑎))
54	53	rspccva 3624	. . . . . . . . . . . . . . . 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 5092	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎)
56	55	adantll 712	. . . . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . 15 ⊢ (((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑏)) = (abs‘𝑦))
58	57	breq1d 5078	. . . . . . . . . . . . . 14 ⊢ (((ℝ D 𝐹)‘𝑏) = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎 ↔ (abs‘𝑦) ≤ 𝑎))
59	56, 58	syl5ibcom 247	. . . . . . . . . . . . 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 3286	. . . . . . . . . . . 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 6733	. . . . . . . . . . . . 13 ⊢ ((Fun (ℝ D 𝐹) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦)
62	10, 61	mpan 688	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → ∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦)
63	60, 62	impel 508	. . . . . . . . . . 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 5071	. . . . . . . . . . 11 ⊢ ((abs‘𝑦) = 𝑏 → ((abs‘𝑦) ≤ 𝑎 ↔ 𝑏 ≤ 𝑎))
65	63, 64	syl5ibcom 247	. . . . . . . . . 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 3286	. . . . . . . . 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 34	. . . . . . . 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 3183	. . . . . . 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 415	. . . . . 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 3276	. . . . 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 15	. . . 4 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎)
72	7, 39, 71	suprcld 11606	. . 3 ⊢ (𝜑 → sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ)
73	1, 72	eqeltrid 2919	. 2 ⊢ (𝜑 → 𝐾 ∈ ℝ)
74		simplrr 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝐴[,]𝐵))
75	74	fvresd 6692	. . . . . . . . . 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 23503	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
78	76, 77	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
79	78	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
80	79, 74	ffvelrnd 6854	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
81	80	recnd 10671	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℂ)
82	75, 81	eqeltrrd 2916	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑦) ∈ ℂ)
83		simplrl 775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝐴[,]𝐵))
84	83	fvresd 6692	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
85	79, 83	ffvelrnd 6854	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
86	85	recnd 10671	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℂ)
87	84, 86	eqeltrrd 2916	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) ∈ ℂ)
88	82, 87	subcld 10999	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑦) − (𝐹‘𝑥)) ∈ ℂ)
89		iccssre 12821	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
90	18, 20, 89	syl2anc 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
91	90	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ ℝ)
92	91, 74	sseldd 3970	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
93	91, 83	sseldd 3970	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
94	92, 93	resubcld 11070	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ)
95	94	recnd 10671	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℂ)
96		simpr 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
97		difrp 12430	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℝ⁺))
98	93, 92, 97	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℝ⁺))
99	96, 98	mpbid 234	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ⁺)
100	99	rpne0d 12439	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ≠ 0)
101	88, 95, 100	absdivd 14817	. . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
104	71	ad2antrr 724	. . . . . . . . 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 11418	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ℂ)
107	106, 34	eleqtrrdi 2926	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ dom abs)
108	93	rexrd 10693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ^*)
109	92	rexrd 10693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ^*)
110	93, 92, 96	ltled 10790	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦)
111		ubicc2 12856	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ (𝑥[,]𝑦))
112	108, 109, 110, 111	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝑥[,]𝑦))
113	112	fvresd 6692	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) = (𝐹‘𝑦))
114		lbicc2 12855	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑥 ≤ 𝑦) → 𝑥 ∈ (𝑥[,]𝑦))
115	108, 109, 110, 114	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝑥[,]𝑦))
116	115	fvresd 6692	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥) = (𝐹‘𝑥))
117	113, 116	oveq12d 7176	. . . . . . . . . . . 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 7173	. . . . . . . . . . 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 12810	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥[,]𝑦) ⊆ (𝐴[,]𝐵))
120	119	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥[,]𝑦) ⊆ (𝐴[,]𝐵))
121	120	resabs1d 5886	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) = (𝐹 ↾ (𝑥[,]𝑦)))
122	76	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
123		rescncf 23507	. . . . . . . . . . . . . . 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 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ))
125	121, 124	eqeltrrd 2916	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹 ∈ (ℂ ↑_pm ℝ))
129		cnex 10620	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ ∈ V
130		reex 10630	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ∈ V
131	129, 130	elpm2 8440	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
132	131	simplbi 500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
133	128, 132	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹:dom 𝐹⟶ℂ)
134	131	simprbi 499	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → dom 𝐹 ⊆ ℝ)
135	128, 134	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom 𝐹 ⊆ ℝ)
136		iccssre 12821	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥[,]𝑦) ⊆ ℝ)
137	93, 92, 136	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥[,]𝑦) ⊆ ℝ)
138		eqid 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
139	138	tgioo2 23413	. . . . . . . . . . . . . . . . . 18 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
140	138, 139	dvres 24511	. . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . 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 23431	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
143	93, 92, 142	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
144	143	reseq2d 5855	. . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
146	145	dmeqd 5776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
147		ioossicc 12825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(,)𝑦) ⊆ (𝑥[,]𝑦)
148	147, 120	sstrid 3980	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥(,)𝑦) ⊆ (𝐴[,]𝐵))
149	17	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
150	148, 149	sstrd 3979	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥(,)𝑦) ⊆ dom (ℝ D 𝐹))
151		ssdmres 5878	. . . . . . . . . . . . . . 15 ⊢ ((𝑥(,)𝑦) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)) = (𝑥(,)𝑦))
152	150, 151	sylib 220	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)) = (𝑥(,)𝑦))
153	146, 152	eqtrd 2858	. . . . . . . . . . . . 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 24591	. . . . . . . . . . . 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 6674	. . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . 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 6691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑥(,)𝑦) → (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
158	157	ad2antll 727	. . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
162	148	sseld 3968	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑎 ∈ (𝑥(,)𝑦) → 𝑎 ∈ (𝐴[,]𝐵)))
163	162	impr 457	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → 𝑎 ∈ (𝐴[,]𝐵))
164		funfvima2 6995	. . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . 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 2915	. . . . . . . . . . . . . . 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 2902	. . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . 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 459	. . . . . . . . . . . . 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 3285	. . . . . . . . . . . 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 15	. . . . . . . . . . 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 2916	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
174		funfvima 6994	. . . . . . . . . . 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 409	. . . . . . . . . 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 835	. . . . . . . . 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 11605	. . . . . . . 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 5110	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ≤ 𝐾)
179	101, 178	eqbrtrrd 5092	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (abs‘(𝑦 − 𝑥))) ≤ 𝐾)
180	88	abscld 14798	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ∈ ℝ)
181	73	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℝ)
182	95, 100	absrpcld 14810	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(𝑦 − 𝑥)) ∈ ℝ⁺)
183	180, 181, 182	ledivmuld 12487	. . . . . 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 234	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ ((abs‘(𝑦 − 𝑥)) · 𝐾))
185	182	rpcnd 12436	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(𝑦 − 𝑥)) ∈ ℂ)
186	181	recnd 10671	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℂ)
187	185, 186	mulcomd 10664	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((abs‘(𝑦 − 𝑥)) · 𝐾) = (𝐾 · (abs‘(𝑦 − 𝑥))))
188	184, 187	breqtrd 5094	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥))))
189	188	ex 415	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥)))))
190	189	ralrimivva 3193	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥)))))
191	73, 190	jca 514	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

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 dom cdm 5557 ran crn 5558 ↾ cres 5559 “ cima 5560 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑_pm cpm 8409 supcsup 8906 ℂcc 10537 ℝcr 10538 · cmul 10544 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℝ⁺crp 12392 (,)cioo 12741 [,]cicc 12744 abscabs 14595 TopOpenctopn 16697 topGenctg 16713 ℂ_fldccnfld 20547 intcnt 21627 –cn→ccncf 23486 D cdv 24463

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467

This theorem is referenced by: c1lip1 24596

W3C validator