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Theorem c1lip1 23979
Description: C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Hypotheses
Ref Expression
c1lip1.a (𝜑𝐴 ∈ ℝ)
c1lip1.b (𝜑𝐵 ∈ ℝ)
c1lip1.f (𝜑𝐹 ∈ (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‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
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.
StepHypRef Expression
1 0re 10252 . . . 4 0 ∈ ℝ
21ne0ii 4066 . . 3 ℝ ≠ ∅
3 ral0 4220 . . . . 5 𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))
4 c1lip1.a . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
54rexrd 10301 . . . . . . . 8 (𝜑𝐴 ∈ ℝ*)
6 c1lip1.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ)
76rexrd 10301 . . . . . . . 8 (𝜑𝐵 ∈ ℝ*)
8 icc0 12436 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
95, 7, 8syl2anc 696 . . . . . . 7 (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
109biimpar 503 . . . . . 6 ((𝜑𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
1110raleqdv 3283 . . . . 5 ((𝜑𝐵 < 𝐴) → (∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))) ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
123, 11mpbiri 248 . . . 4 ((𝜑𝐵 < 𝐴) → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
1312ralrimivw 3105 . . 3 ((𝜑𝐵 < 𝐴) → ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
14 r19.2z 4204 . . 3 ((ℝ ≠ ∅ ∧ ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
152, 13, 14sylancr 698 . 2 ((𝜑𝐵 < 𝐴) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
164adantr 472 . . . . 5 ((𝜑𝐴𝐵) → 𝐴 ∈ ℝ)
176adantr 472 . . . . 5 ((𝜑𝐴𝐵) → 𝐵 ∈ ℝ)
18 simpr 479 . . . . 5 ((𝜑𝐴𝐵) → 𝐴𝐵)
19 c1lip1.f . . . . . 6 (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
2019adantr 472 . . . . 5 ((𝜑𝐴𝐵) → 𝐹 ∈ (ℂ ↑pm ℝ))
21 c1lip1.dv . . . . . 6 (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
2221adantr 472 . . . . 5 ((𝜑𝐴𝐵) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
23 c1lip1.cn . . . . . 6 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
2423adantr 472 . . . . 5 ((𝜑𝐴𝐵) → (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, < )
2616, 17, 18, 20, 22, 24, 25c1liplem1 23978 . . . 4 ((𝜑𝐴𝐵) → (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ ∧ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏𝑎))))))
27 oveq1 6821 . . . . . . . 8 (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → (𝑘 · (abs‘(𝑏𝑎))) = (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏𝑎))))
2827breq2d 4816 . . . . . . 7 (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → ((abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎))) ↔ (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏𝑎)))))
2928imbi2d 329 . . . . . 6 (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → ((𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) ↔ (𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏𝑎))))))
30292ralbidv 3127 . . . . 5 (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) ↔ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏𝑎))))))
3130rspcev 3449 . . . 4 ((sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ ∧ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏𝑎))))) → ∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))))
3226, 31syl 17 . . 3 ((𝜑𝐴𝐵) → ∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))))
33 breq1 4807 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎 < 𝑏𝑥 < 𝑏))
34 fveq2 6353 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
3534oveq2d 6830 . . . . . . . . . . . 12 (𝑎 = 𝑥 → ((𝐹𝑏) − (𝐹𝑎)) = ((𝐹𝑏) − (𝐹𝑥)))
3635fveq2d 6357 . . . . . . . . . . 11 (𝑎 = 𝑥 → (abs‘((𝐹𝑏) − (𝐹𝑎))) = (abs‘((𝐹𝑏) − (𝐹𝑥))))
37 oveq2 6822 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑏𝑎) = (𝑏𝑥))
3837fveq2d 6357 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (abs‘(𝑏𝑎)) = (abs‘(𝑏𝑥)))
3938oveq2d 6830 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑘 · (abs‘(𝑏𝑎))) = (𝑘 · (abs‘(𝑏𝑥))))
4036, 39breq12d 4817 . . . . . . . . . 10 (𝑎 = 𝑥 → ((abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎))) ↔ (abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥)))))
4133, 40imbi12d 333 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) ↔ (𝑥 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥))))))
42 breq2 4808 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝑥 < 𝑏𝑥 < 𝑦))
43 fveq2 6353 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝐹𝑏) = (𝐹𝑦))
4443oveq1d 6829 . . . . . . . . . . . 12 (𝑏 = 𝑦 → ((𝐹𝑏) − (𝐹𝑥)) = ((𝐹𝑦) − (𝐹𝑥)))
4544fveq2d 6357 . . . . . . . . . . 11 (𝑏 = 𝑦 → (abs‘((𝐹𝑏) − (𝐹𝑥))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
46 oveq1 6821 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏𝑥) = (𝑦𝑥))
4746fveq2d 6357 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (abs‘(𝑏𝑥)) = (abs‘(𝑦𝑥)))
4847oveq2d 6830 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑘 · (abs‘(𝑏𝑥))) = (𝑘 · (abs‘(𝑦𝑥))))
4945, 48breq12d 4817 . . . . . . . . . 10 (𝑏 = 𝑦 → ((abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥))) ↔ (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
5042, 49imbi12d 333 . . . . . . . . 9 (𝑏 = 𝑦 → ((𝑥 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥)))) ↔ (𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))))
5141, 50rspc2v 3461 . . . . . . . 8 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))))
5251ad2antlr 765 . . . . . . 7 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))))
53 pm2.27 42 . . . . . . . 8 (𝑥 < 𝑦 → ((𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
5453adantl 473 . . . . . . 7 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
5552, 54syld 47 . . . . . 6 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
56 0le0 11322 . . . . . . . . . 10 0 ≤ 0
57 fvres 6369 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹𝑥))
5857ad2antrl 766 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹𝑥))
59 cncff 22917 . . . . . . . . . . . . . . . . . 18 ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
6023, 59syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
6160ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
62 simpl 474 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
63 ffvelrn 6521 . . . . . . . . . . . . . . . 16 (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
6461, 62, 63syl2an 495 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
6558, 64eqeltrrd 2840 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑥) ∈ ℝ)
6665recnd 10280 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑥) ∈ ℂ)
6766subidd 10592 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹𝑥) − (𝐹𝑥)) = 0)
6867abs00bd 14250 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑥) − (𝐹𝑥))) = 0)
69 iccssre 12468 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
704, 6, 69syl2anc 696 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
7170ad3antrrr 768 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐴[,]𝐵) ⊆ ℝ)
72 simprl 811 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐴[,]𝐵))
7371, 72sseldd 3745 . . . . . . . . . . . . . . . 16 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ)
7473recnd 10280 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℂ)
7574subidd 10592 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥𝑥) = 0)
7675abs00bd 14250 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥𝑥)) = 0)
7776oveq2d 6830 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥𝑥))) = (𝑘 · 0))
78 simplr 809 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℝ)
7978recnd 10280 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℂ)
8079mul01d 10447 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · 0) = 0)
8177, 80eqtrd 2794 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥𝑥))) = 0)
8268, 81breq12d 4817 . . . . . . . . . 10 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((abs‘((𝐹𝑥) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑥𝑥))) ↔ 0 ≤ 0))
8356, 82mpbiri 248 . . . . . . . . 9 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑥) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑥𝑥))))
84 fveq2 6353 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
8584oveq1d 6829 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐹𝑥) − (𝐹𝑥)) = ((𝐹𝑦) − (𝐹𝑥)))
8685fveq2d 6357 . . . . . . . . . 10 (𝑥 = 𝑦 → (abs‘((𝐹𝑥) − (𝐹𝑥))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
87 oveq1 6821 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥𝑥) = (𝑦𝑥))
8887fveq2d 6357 . . . . . . . . . . 11 (𝑥 = 𝑦 → (abs‘(𝑥𝑥)) = (abs‘(𝑦𝑥)))
8988oveq2d 6830 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑘 · (abs‘(𝑥𝑥))) = (𝑘 · (abs‘(𝑦𝑥))))
9086, 89breq12d 4817 . . . . . . . . 9 (𝑥 = 𝑦 → ((abs‘((𝐹𝑥) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑥𝑥))) ↔ (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
9183, 90syl5ibcom 235 . . . . . . . 8 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 = 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
9291imp 444 . . . . . . 7 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 = 𝑦) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
9392a1d 25 . . . . . 6 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 = 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
94 breq1 4807 . . . . . . . . . . 11 (𝑎 = 𝑦 → (𝑎 < 𝑏𝑦 < 𝑏))
95 fveq2 6353 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (𝐹𝑎) = (𝐹𝑦))
9695oveq2d 6830 . . . . . . . . . . . . 13 (𝑎 = 𝑦 → ((𝐹𝑏) − (𝐹𝑎)) = ((𝐹𝑏) − (𝐹𝑦)))
9796fveq2d 6357 . . . . . . . . . . . 12 (𝑎 = 𝑦 → (abs‘((𝐹𝑏) − (𝐹𝑎))) = (abs‘((𝐹𝑏) − (𝐹𝑦))))
98 oveq2 6822 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (𝑏𝑎) = (𝑏𝑦))
9998fveq2d 6357 . . . . . . . . . . . . 13 (𝑎 = 𝑦 → (abs‘(𝑏𝑎)) = (abs‘(𝑏𝑦)))
10099oveq2d 6830 . . . . . . . . . . . 12 (𝑎 = 𝑦 → (𝑘 · (abs‘(𝑏𝑎))) = (𝑘 · (abs‘(𝑏𝑦))))
10197, 100breq12d 4817 . . . . . . . . . . 11 (𝑎 = 𝑦 → ((abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎))) ↔ (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦)))))
10294, 101imbi12d 333 . . . . . . . . . 10 (𝑎 = 𝑦 → ((𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) ↔ (𝑦 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦))))))
103 breq2 4808 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑦 < 𝑏𝑦 < 𝑥))
104 fveq2 6353 . . . . . . . . . . . . . 14 (𝑏 = 𝑥 → (𝐹𝑏) = (𝐹𝑥))
105104oveq1d 6829 . . . . . . . . . . . . 13 (𝑏 = 𝑥 → ((𝐹𝑏) − (𝐹𝑦)) = ((𝐹𝑥) − (𝐹𝑦)))
106105fveq2d 6357 . . . . . . . . . . . 12 (𝑏 = 𝑥 → (abs‘((𝐹𝑏) − (𝐹𝑦))) = (abs‘((𝐹𝑥) − (𝐹𝑦))))
107 oveq1 6821 . . . . . . . . . . . . . 14 (𝑏 = 𝑥 → (𝑏𝑦) = (𝑥𝑦))
108107fveq2d 6357 . . . . . . . . . . . . 13 (𝑏 = 𝑥 → (abs‘(𝑏𝑦)) = (abs‘(𝑥𝑦)))
109108oveq2d 6830 . . . . . . . . . . . 12 (𝑏 = 𝑥 → (𝑘 · (abs‘(𝑏𝑦))) = (𝑘 · (abs‘(𝑥𝑦))))
110106, 109breq12d 4817 . . . . . . . . . . 11 (𝑏 = 𝑥 → ((abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦))) ↔ (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦)))))
111103, 110imbi12d 333 . . . . . . . . . 10 (𝑏 = 𝑥 → ((𝑦 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦)))) ↔ (𝑦 < 𝑥 → (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))))))
112102, 111rspc2v 3461 . . . . . . . . 9 ((𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))))))
113112ancoms 468 . . . . . . . 8 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))))))
114113ad2antlr 765 . . . . . . 7 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))))))
115 simpr 479 . . . . . . . 8 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
116 fvres 6369 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹𝑦))
117116ad2antll 767 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹𝑦))
118 simpr 479 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
119 ffvelrn 6521 . . . . . . . . . . . . . . 15 (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
12061, 118, 119syl2an 495 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
121117, 120eqeltrrd 2840 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑦) ∈ ℝ)
122121recnd 10280 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑦) ∈ ℂ)
12366, 122abssubd 14411 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑥) − (𝐹𝑦))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
124123adantr 472 . . . . . . . . . 10 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘((𝐹𝑥) − (𝐹𝑦))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
12570ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
126125sseld 3743 . . . . . . . . . . . . . . 15 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ))
127125sseld 3743 . . . . . . . . . . . . . . 15 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ))
128126, 127anim12d 587 . . . . . . . . . . . . . 14 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
129128imp 444 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
130 recn 10238 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
131 recn 10238 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
132 abssub 14285 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
133130, 131, 132syl2an 495 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
134129, 133syl 17 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
135134adantr 472 . . . . . . . . . . 11 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
136135oveq2d 6830 . . . . . . . . . 10 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (𝑘 · (abs‘(𝑥𝑦))) = (𝑘 · (abs‘(𝑦𝑥))))
137124, 136breq12d 4817 . . . . . . . . 9 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))) ↔ (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
138137biimpd 219 . . . . . . . 8 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
139115, 138embantd 59 . . . . . . 7 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((𝑦 < 𝑥 → (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦)))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
140114, 139syld 47 . . . . . 6 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
141 lttri4 10334 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
142129, 141syl 17 . . . . . 6 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
14355, 93, 140, 142mpjao3dan 1544 . . . . 5 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
144143ralrimdvva 3112 . . . 4 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
145144reximdva 3155 . . 3 ((𝜑𝐴𝐵) → (∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
14632, 145mpd 15 . 2 ((𝜑𝐴𝐵) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
14715, 146, 6, 4ltlecasei 10357 1 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3o 1071   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  wss 3715  c0 4058   class class class wbr 4804  cres 5268  cima 5269  wf 6045  cfv 6049  (class class class)co 6814  pm cpm 8026  supcsup 8513  cc 10146  cr 10147  0cc0 10148   · cmul 10153  *cxr 10285   < clt 10286  cle 10287  cmin 10478  [,]cicc 12391  abscabs 14193  cnccncf 22900   D cdv 23846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-addf 10227  ax-mulf 10228
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-supp 7465  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fsupp 8443  df-fi 8484  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-q 12002  df-rp 12046  df-xneg 12159  df-xadd 12160  df-xmul 12161  df-ioo 12392  df-ico 12394  df-icc 12395  df-fz 12540  df-fzo 12680  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-mulr 16177  df-starv 16178  df-sca 16179  df-vsca 16180  df-ip 16181  df-tset 16182  df-ple 16183  df-ds 16186  df-unif 16187  df-hom 16188  df-cco 16189  df-rest 16305  df-topn 16306  df-0g 16324  df-gsum 16325  df-topgen 16326  df-pt 16327  df-prds 16330  df-xrs 16384  df-qtop 16389  df-imas 16390  df-xps 16392  df-mre 16468  df-mrc 16469  df-acs 16471  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-submnd 17557  df-mulg 17762  df-cntz 17970  df-cmn 18415  df-psmet 19960  df-xmet 19961  df-met 19962  df-bl 19963  df-mopn 19964  df-fbas 19965  df-fg 19966  df-cnfld 19969  df-top 20921  df-topon 20938  df-topsp 20959  df-bases 20972  df-cld 21045  df-ntr 21046  df-cls 21047  df-nei 21124  df-lp 21162  df-perf 21163  df-cn 21253  df-cnp 21254  df-haus 21341  df-cmp 21412  df-tx 21587  df-hmeo 21780  df-fil 21871  df-fm 21963  df-flim 21964  df-flf 21965  df-xms 22346  df-ms 22347  df-tms 22348  df-cncf 22902  df-limc 23849  df-dv 23850
This theorem is referenced by:  c1lip2  23980
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