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Theorem c1lip1 26121

Description: C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypotheses**
Ref	Expression
c1lip1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
c1lip1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
c1lip1.f	⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
c1lip1.dv	⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
c1lip1.cn	⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))

**Assertion**
Ref	Expression
c1lip1	⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))

Distinct variable groups: 𝜑,𝑥,𝑦,𝑘 𝑥,𝐴,𝑦,𝑘 𝑥,𝐵,𝑦,𝑘 𝑥,𝐹,𝑦,𝑘

Proof of Theorem c1lip1 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 11206	. . . 4 ⊢ 0 ∈ ℝ
2	1	ne0ii 4305	. . 3 ⊢ ℝ ≠ ∅
3		ral0 4461	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))
4		c1lip1.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	rexrd 11255	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
6		c1lip1.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	rexrd 11255	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
8		icc0 13416	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
9	5, 7, 8	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
10	9	biimpar 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
11	10	raleqdv 3329	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))) ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
12	3, 11	mpbiri 261	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
13	12	ralrimivw 3167	. . 3 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
14		r19.2z 4462	. . 3 ⊢ ((ℝ ≠ ∅ ∧ ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
15	2, 13, 14	sylancr 598	. 2 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
16	4	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
17	6	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
18		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
19		c1lip1.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
20	19	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐹 ∈ (ℂ ↑_pm ℝ))
21		c1lip1.dv	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
22	21	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
23		c1lip1.cn	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
24	23	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
25		eqid 2769	. . . . 5 ⊢ sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )
26	16, 17, 18, 20, 22, 24, 25	c1liplem1 26120	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ ∧ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))))
27		oveq1 7415	. . . . . . . 8 ⊢ (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → (𝑘 · (abs‘(𝑏 − 𝑎))) = (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))
28	27	breq2d 5122	. . . . . . 7 ⊢ (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎))) ↔ (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎)))))
29	28	imbi2d 343	. . . . . 6 ⊢ (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → ((𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) ↔ (𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))))
30	29	2ralbidv 3235	. . . . 5 ⊢ (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) ↔ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))))
31	30	rspcev 3590	. . . 4 ⊢ ((sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ ∧ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))) → ∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))))
32	26, 31	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))))
33		breq1 5113	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 < 𝑏 ↔ 𝑥 < 𝑏))
34		fveq2 6879	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
35	34	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑥)))
36	35	fveq2d 6883	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))))
37		oveq2 7416	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑏 − 𝑎) = (𝑏 − 𝑥))
38	37	fveq2d 6883	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑥)))
39	38	oveq2d 7424	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑥))))
40	36, 39	breq12d 5123	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎))) ↔ (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥)))))
41	33, 40	imbi12d 347	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → ((𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) ↔ (𝑥 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥))))))
42		breq2 5114	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑥 < 𝑏 ↔ 𝑥 < 𝑦))
43		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦))
44	43	fvoveq1d 7430	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
45		fvoveq1 7431	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (abs‘(𝑏 − 𝑥)) = (abs‘(𝑦 − 𝑥)))
46	45	oveq2d 7424	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
47	44, 46	breq12d 5123	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
48	42, 47	imbi12d 347	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → ((𝑥 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥)))) ↔ (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))))
49	41, 48	rspc2v 3601	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))))
50	49	ad2antlr 739	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))))
51		pm2.27 43	. . . . . . . 8 ⊢ (𝑥 < 𝑦 → ((𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
52	51	adantl 486	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
53	50, 52	syld 48	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
54		0le0 12338	. . . . . . . . . 10 ⊢ 0 ≤ 0
55		fvres 6898	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
56	55	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
57		cncff 25017	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
58	23, 57	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
59	58	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
60		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
61		ffvelcdm 7074	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
62	59, 60, 61	syl2an 607	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
63	56, 62	eqeltrrd 2870	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℝ)
64	63	recnd 11233	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℂ)
65	64	subidd 11553	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹‘𝑥) − (𝐹‘𝑥)) = 0)
66	65	abs00bd 15338	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = 0)
67		iccssre 13452	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
68	4, 6, 67	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
69	68	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐴[,]𝐵) ⊆ ℝ)
70		simprl 782	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐴[,]𝐵))
71	69, 70	sseldd 3946	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ)
72	71	recnd 11233	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℂ)
73	72	subidd 11553	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 − 𝑥) = 0)
74	73	abs00bd 15338	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥 − 𝑥)) = 0)
75	74	oveq2d 7424	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · 0))
76		simplr 780	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℝ)
77	76	recnd 11233	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℂ)
78	77	mul01d 11405	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · 0) = 0)
79	75, 78	eqtrd 2804	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = 0)
80	66, 79	breq12d 5123	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥))) ↔ 0 ≤ 0))
81	54, 80	mpbiri 261	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥))))
82		fveq2 6879	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
83	82	fvoveq1d 7430	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
84		fvoveq1 7431	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (abs‘(𝑥 − 𝑥)) = (abs‘(𝑦 − 𝑥)))
85	84	oveq2d 7424	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
86	83, 85	breq12d 5123	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
87	81, 86	syl5ibcom 248	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 = 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
88	87	imp 411	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 = 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
89	88	a1d 26	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 = 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
90		breq1 5113	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 < 𝑏 ↔ 𝑦 < 𝑏))
91		fveq2 6879	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
92	91	oveq2d 7424	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑦)))
93	92	fveq2d 6883	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))))
94		oveq2 7416	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (𝑏 − 𝑎) = (𝑏 − 𝑦))
95	94	fveq2d 6883	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑦)))
96	95	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑦))))
97	93, 96	breq12d 5123	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎))) ↔ (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦)))))
98	90, 97	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ((𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) ↔ (𝑦 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦))))))
99		breq2 5114	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑦 < 𝑏 ↔ 𝑦 < 𝑥))
100		fveq2 6879	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (𝐹‘𝑏) = (𝐹‘𝑥))
101	100	fvoveq1d 7430	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))))
102		fvoveq1 7431	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (abs‘(𝑏 − 𝑦)) = (abs‘(𝑥 − 𝑦)))
103	102	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑦))) = (𝑘 · (abs‘(𝑥 − 𝑦))))
104	101, 103	breq12d 5123	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦))) ↔ (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦)))))
105	99, 104	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → ((𝑦 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦)))) ↔ (𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))))
106	98, 105	rspc2v 3601	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))))
107	106	ancoms 463	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))))
108	107	ad2antlr 739	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))))
109		simpr 489	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
110		fvres 6898	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦))
111	110	ad2antll 741	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦))
112		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
113		ffvelcdm 7074	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
114	59, 112, 113	syl2an 607	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
115	111, 114	eqeltrrd 2870	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℝ)
116	115	recnd 11233	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℂ)
117	64, 116	abssubd 15503	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
118	117	adantr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
119	68	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
120	119	sseld 3944	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ))
121	119	sseld 3944	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ))
122	120, 121	anim12d 620	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
123	122	imp 411	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
124		recn 11186	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
125		recn 11186	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
126		abssub 15374	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
127	124, 125, 126	syl2an 607	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
128	123, 127	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
129	128	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
130	129	oveq2d 7424	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (𝑘 · (abs‘(𝑥 − 𝑦))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
131	118, 130	breq12d 5123	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
132	131	biimpd 232	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
133	109, 132	embantd 60	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
134	108, 133	syld 48	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
135		lttri4 11290	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
136	123, 135	syl 18	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
137	53, 89, 134, 136	mpjao3dan 1457	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
138	137	ralrimdvva 3226	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
139	138	reximdva 3184	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
140	32, 139	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
141	15, 140, 6, 4	ltlecasei 11314	1 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ w3o 1100 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ⊆ wss 3913 ∅c0 4294 class class class wbr 5110 ↾ cres 5661 “ cima 5662 ⟶wf 6530 ‘cfv 6534 (class class class)co 7408 ↑_pm cpm 8821 supcsup 9396 ℂcc 11094 ℝcr 11095 0cc0 11096 · cmul 11101 ℝ^*cxr 11238 < clt 11239 ≤ cle 11240 − cmin 11437 [,]cicc 13371 abscabs 15281 –cn→ccncf 25000 D cdv 25987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174 ax-addf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13372 df-ico 13374 df-icc 13375 df-fz 13532 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17471 df-topn 17472 df-0g 17490 df-gsum 17491 df-topgen 17492 df-pt 17493 df-prds 17496 df-xrs 17552 df-qtop 17557 df-imas 17558 df-xps 17560 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-mulg 19130 df-cntz 19383 df-cmn 19848 df-psmet 21479 df-xmet 21480 df-met 21481 df-bl 21482 df-mopn 21483 df-fbas 21484 df-fg 21485 df-cnfld 21488 df-top 23016 df-topon 23033 df-topsp 23055 df-bases 23068 df-cld 23141 df-ntr 23142 df-cls 23143 df-nei 23220 df-lp 23258 df-perf 23259 df-cn 23349 df-cnp 23350 df-haus 23437 df-cmp 23509 df-tx 23684 df-hmeo 23877 df-fil 23968 df-fm 24060 df-flim 24061 df-flf 24062 df-xms 24442 df-ms 24443 df-tms 24444 df-cncf 25002 df-limc 25990 df-dv 25991

This theorem is referenced by: c1lip2 26122

W3C validator