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Theorem c1lip1 23980
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 10242 . . . 4 0 ∈ ℝ
21ne0ii 4071 . . 3 ℝ ≠ ∅
3 ral0 4217 . . . . 5 𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))
4 c1lip1.a . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
54rexrd 10291 . . . . . . . 8 (𝜑𝐴 ∈ ℝ*)
6 c1lip1.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ)
76rexrd 10291 . . . . . . . 8 (𝜑𝐵 ∈ ℝ*)
8 icc0 12428 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
95, 7, 8syl2anc 573 . . . . . . 7 (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
109biimpar 463 . . . . . 6 ((𝜑𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
1110raleqdv 3293 . . . . 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 3116 . . 3 ((𝜑𝐵 < 𝐴) → ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
14 r19.2z 4201 . . 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 575 . 2 ((𝜑𝐵 < 𝐴) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
164adantr 466 . . . . 5 ((𝜑𝐴𝐵) → 𝐴 ∈ ℝ)
176adantr 466 . . . . 5 ((𝜑𝐴𝐵) → 𝐵 ∈ ℝ)
18 simpr 471 . . . . 5 ((𝜑𝐴𝐵) → 𝐴𝐵)
19 c1lip1.f . . . . . 6 (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
2019adantr 466 . . . . 5 ((𝜑𝐴𝐵) → 𝐹 ∈ (ℂ ↑pm ℝ))
21 c1lip1.dv . . . . . 6 (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
2221adantr 466 . . . . 5 ((𝜑𝐴𝐵) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
23 c1lip1.cn . . . . . 6 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
2423adantr 466 . . . . 5 ((𝜑𝐴𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
25 eqid 2771 . . . . 5 sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )
2616, 17, 18, 20, 22, 24, 25c1liplem1 23979 . . . 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 6800 . . . . . . . 8 (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → (𝑘 · (abs‘(𝑏𝑎))) = (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏𝑎))))
2827breq2d 4798 . . . . . . 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 3138 . . . . 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 3460 . . . 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 4789 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎 < 𝑏𝑥 < 𝑏))
34 fveq2 6332 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
3534oveq2d 6809 . . . . . . . . . . . 12 (𝑎 = 𝑥 → ((𝐹𝑏) − (𝐹𝑎)) = ((𝐹𝑏) − (𝐹𝑥)))
3635fveq2d 6336 . . . . . . . . . . 11 (𝑎 = 𝑥 → (abs‘((𝐹𝑏) − (𝐹𝑎))) = (abs‘((𝐹𝑏) − (𝐹𝑥))))
37 oveq2 6801 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑏𝑎) = (𝑏𝑥))
3837fveq2d 6336 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (abs‘(𝑏𝑎)) = (abs‘(𝑏𝑥)))
3938oveq2d 6809 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑘 · (abs‘(𝑏𝑎))) = (𝑘 · (abs‘(𝑏𝑥))))
4036, 39breq12d 4799 . . . . . . . . . 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 4790 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝑥 < 𝑏𝑥 < 𝑦))
43 fveq2 6332 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (𝐹𝑏) = (𝐹𝑦))
4443fvoveq1d 6815 . . . . . . . . . . 11 (𝑏 = 𝑦 → (abs‘((𝐹𝑏) − (𝐹𝑥))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
45 fvoveq1 6816 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (abs‘(𝑏𝑥)) = (abs‘(𝑦𝑥)))
4645oveq2d 6809 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑘 · (abs‘(𝑏𝑥))) = (𝑘 · (abs‘(𝑦𝑥))))
4744, 46breq12d 4799 . . . . . . . . . 10 (𝑏 = 𝑦 → ((abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥))) ↔ (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
4842, 47imbi12d 333 . . . . . . . . 9 (𝑏 = 𝑦 → ((𝑥 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥)))) ↔ (𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))))
4941, 48rspc2v 3472 . . . . . . . 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))))))
5049ad2antlr 706 . . . . . . 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))))))
51 pm2.27 42 . . . . . . . 8 (𝑥 < 𝑦 → ((𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
5251adantl 467 . . . . . . 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)))))
5350, 52syld 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)))))
54 0le0 11312 . . . . . . . . . 10 0 ≤ 0
55 fvres 6348 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹𝑥))
5655ad2antrl 707 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹𝑥))
57 cncff 22916 . . . . . . . . . . . . . . . . . 18 ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
5823, 57syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
5958ad2antrr 705 . . . . . . . . . . . . . . . 16 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
60 simpl 468 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
61 ffvelrn 6500 . . . . . . . . . . . . . . . 16 (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
6259, 60, 61syl2an 583 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
6356, 62eqeltrrd 2851 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑥) ∈ ℝ)
6463recnd 10270 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑥) ∈ ℂ)
6564subidd 10582 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹𝑥) − (𝐹𝑥)) = 0)
6665abs00bd 14239 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑥) − (𝐹𝑥))) = 0)
67 iccssre 12460 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
684, 6, 67syl2anc 573 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
6968ad3antrrr 709 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐴[,]𝐵) ⊆ ℝ)
70 simprl 754 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐴[,]𝐵))
7169, 70sseldd 3753 . . . . . . . . . . . . . . . 16 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ)
7271recnd 10270 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℂ)
7372subidd 10582 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥𝑥) = 0)
7473abs00bd 14239 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥𝑥)) = 0)
7574oveq2d 6809 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥𝑥))) = (𝑘 · 0))
76 simplr 752 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℝ)
7776recnd 10270 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℂ)
7877mul01d 10437 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · 0) = 0)
7975, 78eqtrd 2805 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥𝑥))) = 0)
8066, 79breq12d 4799 . . . . . . . . . 10 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((abs‘((𝐹𝑥) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑥𝑥))) ↔ 0 ≤ 0))
8154, 80mpbiri 248 . . . . . . . . 9 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑥) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑥𝑥))))
82 fveq2 6332 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
8382fvoveq1d 6815 . . . . . . . . . 10 (𝑥 = 𝑦 → (abs‘((𝐹𝑥) − (𝐹𝑥))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
84 fvoveq1 6816 . . . . . . . . . . 11 (𝑥 = 𝑦 → (abs‘(𝑥𝑥)) = (abs‘(𝑦𝑥)))
8584oveq2d 6809 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑘 · (abs‘(𝑥𝑥))) = (𝑘 · (abs‘(𝑦𝑥))))
8683, 85breq12d 4799 . . . . . . . . 9 (𝑥 = 𝑦 → ((abs‘((𝐹𝑥) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑥𝑥))) ↔ (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
8781, 86syl5ibcom 235 . . . . . . . 8 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 = 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
8887imp 393 . . . . . . 7 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 = 𝑦) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
8988a1d 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)))))
90 breq1 4789 . . . . . . . . . . 11 (𝑎 = 𝑦 → (𝑎 < 𝑏𝑦 < 𝑏))
91 fveq2 6332 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (𝐹𝑎) = (𝐹𝑦))
9291oveq2d 6809 . . . . . . . . . . . . 13 (𝑎 = 𝑦 → ((𝐹𝑏) − (𝐹𝑎)) = ((𝐹𝑏) − (𝐹𝑦)))
9392fveq2d 6336 . . . . . . . . . . . 12 (𝑎 = 𝑦 → (abs‘((𝐹𝑏) − (𝐹𝑎))) = (abs‘((𝐹𝑏) − (𝐹𝑦))))
94 oveq2 6801 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (𝑏𝑎) = (𝑏𝑦))
9594fveq2d 6336 . . . . . . . . . . . . 13 (𝑎 = 𝑦 → (abs‘(𝑏𝑎)) = (abs‘(𝑏𝑦)))
9695oveq2d 6809 . . . . . . . . . . . 12 (𝑎 = 𝑦 → (𝑘 · (abs‘(𝑏𝑎))) = (𝑘 · (abs‘(𝑏𝑦))))
9793, 96breq12d 4799 . . . . . . . . . . 11 (𝑎 = 𝑦 → ((abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎))) ↔ (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦)))))
9890, 97imbi12d 333 . . . . . . . . . 10 (𝑎 = 𝑦 → ((𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) ↔ (𝑦 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦))))))
99 breq2 4790 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑦 < 𝑏𝑦 < 𝑥))
100 fveq2 6332 . . . . . . . . . . . . 13 (𝑏 = 𝑥 → (𝐹𝑏) = (𝐹𝑥))
101100fvoveq1d 6815 . . . . . . . . . . . 12 (𝑏 = 𝑥 → (abs‘((𝐹𝑏) − (𝐹𝑦))) = (abs‘((𝐹𝑥) − (𝐹𝑦))))
102 fvoveq1 6816 . . . . . . . . . . . . 13 (𝑏 = 𝑥 → (abs‘(𝑏𝑦)) = (abs‘(𝑥𝑦)))
103102oveq2d 6809 . . . . . . . . . . . 12 (𝑏 = 𝑥 → (𝑘 · (abs‘(𝑏𝑦))) = (𝑘 · (abs‘(𝑥𝑦))))
104101, 103breq12d 4799 . . . . . . . . . . 11 (𝑏 = 𝑥 → ((abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦))) ↔ (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦)))))
10599, 104imbi12d 333 . . . . . . . . . 10 (𝑏 = 𝑥 → ((𝑦 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦)))) ↔ (𝑦 < 𝑥 → (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))))))
10698, 105rspc2v 3472 . . . . . . . . 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))))))
107106ancoms 455 . . . . . . . 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))))))
108107ad2antlr 706 . . . . . . 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))))))
109 simpr 471 . . . . . . . 8 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
110 fvres 6348 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹𝑦))
111110ad2antll 708 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹𝑦))
112 simpr 471 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
113 ffvelrn 6500 . . . . . . . . . . . . . . 15 (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
11459, 112, 113syl2an 583 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
115111, 114eqeltrrd 2851 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑦) ∈ ℝ)
116115recnd 10270 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑦) ∈ ℂ)
11764, 116abssubd 14400 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑥) − (𝐹𝑦))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
118117adantr 466 . . . . . . . . . 10 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘((𝐹𝑥) − (𝐹𝑦))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
11968ad2antrr 705 . . . . . . . . . . . . . . . 16 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
120119sseld 3751 . . . . . . . . . . . . . . 15 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ))
121119sseld 3751 . . . . . . . . . . . . . . 15 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ))
122120, 121anim12d 596 . . . . . . . . . . . . . 14 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
123122imp 393 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
124 recn 10228 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
125 recn 10228 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
126 abssub 14274 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
127124, 125, 126syl2an 583 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
128123, 127syl 17 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
129128adantr 466 . . . . . . . . . . 11 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
130129oveq2d 6809 . . . . . . . . . 10 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (𝑘 · (abs‘(𝑥𝑦))) = (𝑘 · (abs‘(𝑦𝑥))))
131118, 130breq12d 4799 . . . . . . . . 9 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))) ↔ (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
132131biimpd 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)))))
133109, 132embantd 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)))))
134108, 133syld 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)))))
135 lttri4 10324 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
136123, 135syl 17 . . . . . 6 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
13753, 89, 134, 136mpjao3dan 1543 . . . . 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)))))
138137ralrimdvva 3123 . . . 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)))))
139138reximdva 3165 . . 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)))))
14032, 139mpd 15 . 2 ((𝜑𝐴𝐵) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
14115, 140, 6, 4ltlecasei 10347 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 382  w3o 1070   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  wss 3723  c0 4063   class class class wbr 4786  cres 5251  cima 5252  wf 6027  cfv 6031  (class class class)co 6793  pm cpm 8010  supcsup 8502  cc 10136  cr 10137  0cc0 10138   · cmul 10143  *cxr 10275   < clt 10276  cle 10277  cmin 10468  [,]cicc 12383  abscabs 14182  cnccncf 22899   D cdv 23847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-addf 10217  ax-mulf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-hom 16174  df-cco 16175  df-rest 16291  df-topn 16292  df-0g 16310  df-gsum 16311  df-topgen 16312  df-pt 16313  df-prds 16316  df-xrs 16370  df-qtop 16375  df-imas 16376  df-xps 16378  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-mulg 17749  df-cntz 17957  df-cmn 18402  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-fbas 19958  df-fg 19959  df-cnfld 19962  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-lp 21161  df-perf 21162  df-cn 21252  df-cnp 21253  df-haus 21340  df-cmp 21411  df-tx 21586  df-hmeo 21779  df-fil 21870  df-fm 21962  df-flim 21963  df-flf 21964  df-xms 22345  df-ms 22346  df-tms 22347  df-cncf 22901  df-limc 23850  df-dv 23851
This theorem is referenced by:  c1lip2  23981
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