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Theorem c1lip1 26326

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 11317	. . . 4 ⊢ 0 ∈ ℝ
2	1	ne0ii 4289	. . 3 ⊢ ℝ ≠ ∅
3		ral0 4453	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))
4		c1lip1.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	rexrd 11366	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
6		c1lip1.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	rexrd 11366	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
8		icc0 13531	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
9	5, 7, 8	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
10	9	biimpar 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
11	10	raleqdv 3319	. . . . 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 3158	. . 3 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
14		r19.2z 4454	. . 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 599	. 2 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
16	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
17	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
18		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
19		c1lip1.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
20	19	adantr 486	. . . . 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 486	. . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
25		eqid 2760	. . . . 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 26325	. . . 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 7431	. . . . . . . 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 5114	. . . . . . 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 3226	. . . . 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 3576	. . . 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 5105	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 < 𝑏 ↔ 𝑥 < 𝑏))
34		fveq2 6890	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
35	34	oveq2d 7440	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑥)))
36	35	fveq2d 6894	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))))
37		oveq2 7432	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑏 − 𝑎) = (𝑏 − 𝑥))
38	37	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑥)))
39	38	oveq2d 7440	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑥))))
40	36, 39	breq12d 5115	. . . . . . . . . 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 5106	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑥 < 𝑏 ↔ 𝑥 < 𝑦))
43		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦))
44	43	fvoveq1d 7446	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
45		fvoveq1 7447	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (abs‘(𝑏 − 𝑥)) = (abs‘(𝑦 − 𝑥)))
46	45	oveq2d 7440	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
47	44, 46	breq12d 5115	. . . . . . . . . 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 3586	. . . . . . . 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 740	. . . . . . 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 487	. . . . . . 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 12451	. . . . . . . . . 10 ⊢ 0 ≤ 0
55		fvres 6909	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
56	55	ad2antrl 741	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
57		cncff 25223	. . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
60		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
61		ffvelcdm 7086	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
62	59, 60, 61	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
63	56, 62	eqeltrrd 2861	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℝ)
64	63	recnd 11344	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℂ)
65	64	subidd 11664	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹‘𝑥) − (𝐹‘𝑥)) = 0)
66	65	abs00bd 15467	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = 0)
67		iccssre 13567	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
68	4, 6, 67	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
69	68	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐴[,]𝐵) ⊆ ℝ)
70		simprl 783	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐴[,]𝐵))
71	69, 70	sseldd 3910	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ)
72	71	recnd 11344	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℂ)
73	72	subidd 11664	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 − 𝑥) = 0)
74	73	abs00bd 15467	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥 − 𝑥)) = 0)
75	74	oveq2d 7440	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · 0))
76		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℝ)
77	76	recnd 11344	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℂ)
78	77	mul01d 11516	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · 0) = 0)
79	75, 78	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = 0)
80	66, 79	breq12d 5115	. . . . . . . . . 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 6890	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
83	82	fvoveq1d 7446	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
84		fvoveq1 7447	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (abs‘(𝑥 − 𝑥)) = (abs‘(𝑦 − 𝑥)))
85	84	oveq2d 7440	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
86	83, 85	breq12d 5115	. . . . . . . . 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 412	. . . . . . 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 5105	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 < 𝑏 ↔ 𝑦 < 𝑏))
91		fveq2 6890	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
92	91	oveq2d 7440	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑦)))
93	92	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))))
94		oveq2 7432	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (𝑏 − 𝑎) = (𝑏 − 𝑦))
95	94	fveq2d 6894	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑦)))
96	95	oveq2d 7440	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑦))))
97	93, 96	breq12d 5115	. . . . . . . . . . 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 5106	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑦 < 𝑏 ↔ 𝑦 < 𝑥))
100		fveq2 6890	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (𝐹‘𝑏) = (𝐹‘𝑥))
101	100	fvoveq1d 7446	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))))
102		fvoveq1 7447	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (abs‘(𝑏 − 𝑦)) = (abs‘(𝑥 − 𝑦)))
103	102	oveq2d 7440	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑦))) = (𝑘 · (abs‘(𝑥 − 𝑦))))
104	101, 103	breq12d 5115	. . . . . . . . . . 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 3586	. . . . . . . . 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 464	. . . . . . . 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 740	. . . . . . 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 490	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
110		fvres 6909	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦))
111	110	ad2antll 742	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦))
112		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
113		ffvelcdm 7086	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
114	59, 112, 113	syl2an 608	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
115	111, 114	eqeltrrd 2861	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℝ)
116	115	recnd 11344	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℂ)
117	64, 116	abssubd 15632	. . . . . . . . . . 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 486	. . . . . . . . . 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 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
120	119	sseld 3908	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ))
121	119	sseld 3908	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ))
122	120, 121	anim12d 621	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
123	122	imp 412	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
124		recn 11297	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
125		recn 11297	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
126		abssub 15503	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
127	124, 125, 126	syl2an 608	. . . . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
130	129	oveq2d 7440	. . . . . . . . . 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 5115	. . . . . . . . 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 11401	. . . . . . 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 1459	. . . . 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 3217	. . . 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 3175	. . 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 11425	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3896 ∅c0 4277 class class class wbr 5102 ↾ cres 5653 “ cima 5654 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ↑_pm cpm 8855 supcsup 9439 ℂcc 11205 ℝcr 11206 0cc0 11207 · cmul 11212 ℝ^*cxr 11349 < clt 11350 ≤ cle 11351 − cmin 11548 [,]cicc 13486 abscabs 15410 –cn→ccncf 25206 D cdv 26192

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 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 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8184 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-2o 8484 df-er 8724 df-map 8856 df-pm 8857 df-ixp 8933 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-fsupp 9361 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-cmp 23714 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196

This theorem is used by: c1lip2 26327

W3C validator