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Theorem c1lip1 23677
Description: C1 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 9991 . . . 4 0 ∈ ℝ
21ne0ii 3904 . . 3 ℝ ≠ ∅
3 ral0 4053 . . . . 5 𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))
4 c1lip1.a . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
54rexrd 10040 . . . . . . . 8 (𝜑𝐴 ∈ ℝ*)
6 c1lip1.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ)
76rexrd 10040 . . . . . . . 8 (𝜑𝐵 ∈ ℝ*)
8 icc0 12172 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
95, 7, 8syl2anc 692 . . . . . . 7 (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
109biimpar 502 . . . . . 6 ((𝜑𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
1110raleqdv 3136 . . . . 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 2962 . . 3 ((𝜑𝐵 < 𝐴) → ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
14 r19.2z 4037 . . 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 694 . 2 ((𝜑𝐵 < 𝐴) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))
164adantr 481 . . . . 5 ((𝜑𝐴𝐵) → 𝐴 ∈ ℝ)
176adantr 481 . . . . 5 ((𝜑𝐴𝐵) → 𝐵 ∈ ℝ)
18 simpr 477 . . . . 5 ((𝜑𝐴𝐵) → 𝐴𝐵)
19 c1lip1.f . . . . . 6 (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
2019adantr 481 . . . . 5 ((𝜑𝐴𝐵) → 𝐹 ∈ (ℂ ↑pm ℝ))
21 c1lip1.dv . . . . . 6 (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
2221adantr 481 . . . . 5 ((𝜑𝐴𝐵) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
23 c1lip1.cn . . . . . 6 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
2423adantr 481 . . . . 5 ((𝜑𝐴𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
25 eqid 2621 . . . . 5 sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )
2616, 17, 18, 20, 22, 24, 25c1liplem1 23676 . . . 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 6617 . . . . . . . 8 (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → (𝑘 · (abs‘(𝑏𝑎))) = (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏𝑎))))
2827breq2d 4630 . . . . . . 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 330 . . . . . 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 2984 . . . . 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 3298 . . . 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 4621 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎 < 𝑏𝑥 < 𝑏))
34 fveq2 6153 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
3534oveq2d 6626 . . . . . . . . . . . 12 (𝑎 = 𝑥 → ((𝐹𝑏) − (𝐹𝑎)) = ((𝐹𝑏) − (𝐹𝑥)))
3635fveq2d 6157 . . . . . . . . . . 11 (𝑎 = 𝑥 → (abs‘((𝐹𝑏) − (𝐹𝑎))) = (abs‘((𝐹𝑏) − (𝐹𝑥))))
37 oveq2 6618 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑏𝑎) = (𝑏𝑥))
3837fveq2d 6157 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (abs‘(𝑏𝑎)) = (abs‘(𝑏𝑥)))
3938oveq2d 6626 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑘 · (abs‘(𝑏𝑎))) = (𝑘 · (abs‘(𝑏𝑥))))
4036, 39breq12d 4631 . . . . . . . . . 10 (𝑎 = 𝑥 → ((abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎))) ↔ (abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥)))))
4133, 40imbi12d 334 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) ↔ (𝑥 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥))))))
42 breq2 4622 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝑥 < 𝑏𝑥 < 𝑦))
43 fveq2 6153 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝐹𝑏) = (𝐹𝑦))
4443oveq1d 6625 . . . . . . . . . . . 12 (𝑏 = 𝑦 → ((𝐹𝑏) − (𝐹𝑥)) = ((𝐹𝑦) − (𝐹𝑥)))
4544fveq2d 6157 . . . . . . . . . . 11 (𝑏 = 𝑦 → (abs‘((𝐹𝑏) − (𝐹𝑥))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
46 oveq1 6617 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏𝑥) = (𝑦𝑥))
4746fveq2d 6157 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (abs‘(𝑏𝑥)) = (abs‘(𝑦𝑥)))
4847oveq2d 6626 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑘 · (abs‘(𝑏𝑥))) = (𝑘 · (abs‘(𝑦𝑥))))
4945, 48breq12d 4631 . . . . . . . . . 10 (𝑏 = 𝑦 → ((abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥))) ↔ (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥)))))
5042, 49imbi12d 334 . . . . . . . . 9 (𝑏 = 𝑦 → ((𝑥 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑏𝑥)))) ↔ (𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))))
5141, 50rspc2v 3310 . . . . . . . 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 762 . . . . . . 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 482 . . . . . . 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 11061 . . . . . . . . . 10 0 ≤ 0
57 fvres 6169 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹𝑥))
5857ad2antrl 763 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹𝑥))
59 cncff 22615 . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . 16 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
62 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
63 ffvelrn 6318 . . . . . . . . . . . . . . . 16 (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
6461, 62, 63syl2an 494 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
6558, 64eqeltrrd 2699 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑥) ∈ ℝ)
6665recnd 10019 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑥) ∈ ℂ)
6766subidd 10331 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹𝑥) − (𝐹𝑥)) = 0)
6867abs00bd 13972 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑥) − (𝐹𝑥))) = 0)
69 iccssre 12204 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
704, 6, 69syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
7170ad3antrrr 765 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐴[,]𝐵) ⊆ ℝ)
72 simprl 793 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐴[,]𝐵))
7371, 72sseldd 3588 . . . . . . . . . . . . . . . 16 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ)
7473recnd 10019 . . . . . . . . . . . . . . 15 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℂ)
7574subidd 10331 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥𝑥) = 0)
7675abs00bd 13972 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥𝑥)) = 0)
7776oveq2d 6626 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥𝑥))) = (𝑘 · 0))
78 simplr 791 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℝ)
7978recnd 10019 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℂ)
8079mul01d 10186 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · 0) = 0)
8177, 80eqtrd 2655 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥𝑥))) = 0)
8268, 81breq12d 4631 . . . . . . . . . 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 6153 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
8584oveq1d 6625 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝐹𝑥) − (𝐹𝑥)) = ((𝐹𝑦) − (𝐹𝑥)))
8685fveq2d 6157 . . . . . . . . . 10 (𝑥 = 𝑦 → (abs‘((𝐹𝑥) − (𝐹𝑥))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
87 oveq1 6617 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥𝑥) = (𝑦𝑥))
8887fveq2d 6157 . . . . . . . . . . 11 (𝑥 = 𝑦 → (abs‘(𝑥𝑥)) = (abs‘(𝑦𝑥)))
8988oveq2d 6626 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑘 · (abs‘(𝑥𝑥))) = (𝑘 · (abs‘(𝑦𝑥))))
9086, 89breq12d 4631 . . . . . . . . 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 445 . . . . . . 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 4621 . . . . . . . . . . 11 (𝑎 = 𝑦 → (𝑎 < 𝑏𝑦 < 𝑏))
95 fveq2 6153 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (𝐹𝑎) = (𝐹𝑦))
9695oveq2d 6626 . . . . . . . . . . . . 13 (𝑎 = 𝑦 → ((𝐹𝑏) − (𝐹𝑎)) = ((𝐹𝑏) − (𝐹𝑦)))
9796fveq2d 6157 . . . . . . . . . . . 12 (𝑎 = 𝑦 → (abs‘((𝐹𝑏) − (𝐹𝑎))) = (abs‘((𝐹𝑏) − (𝐹𝑦))))
98 oveq2 6618 . . . . . . . . . . . . . 14 (𝑎 = 𝑦 → (𝑏𝑎) = (𝑏𝑦))
9998fveq2d 6157 . . . . . . . . . . . . 13 (𝑎 = 𝑦 → (abs‘(𝑏𝑎)) = (abs‘(𝑏𝑦)))
10099oveq2d 6626 . . . . . . . . . . . 12 (𝑎 = 𝑦 → (𝑘 · (abs‘(𝑏𝑎))) = (𝑘 · (abs‘(𝑏𝑦))))
10197, 100breq12d 4631 . . . . . . . . . . 11 (𝑎 = 𝑦 → ((abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎))) ↔ (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦)))))
10294, 101imbi12d 334 . . . . . . . . . 10 (𝑎 = 𝑦 → ((𝑎 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑎))) ≤ (𝑘 · (abs‘(𝑏𝑎)))) ↔ (𝑦 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦))))))
103 breq2 4622 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑦 < 𝑏𝑦 < 𝑥))
104 fveq2 6153 . . . . . . . . . . . . . 14 (𝑏 = 𝑥 → (𝐹𝑏) = (𝐹𝑥))
105104oveq1d 6625 . . . . . . . . . . . . 13 (𝑏 = 𝑥 → ((𝐹𝑏) − (𝐹𝑦)) = ((𝐹𝑥) − (𝐹𝑦)))
106105fveq2d 6157 . . . . . . . . . . . 12 (𝑏 = 𝑥 → (abs‘((𝐹𝑏) − (𝐹𝑦))) = (abs‘((𝐹𝑥) − (𝐹𝑦))))
107 oveq1 6617 . . . . . . . . . . . . . 14 (𝑏 = 𝑥 → (𝑏𝑦) = (𝑥𝑦))
108107fveq2d 6157 . . . . . . . . . . . . 13 (𝑏 = 𝑥 → (abs‘(𝑏𝑦)) = (abs‘(𝑥𝑦)))
109108oveq2d 6626 . . . . . . . . . . . 12 (𝑏 = 𝑥 → (𝑘 · (abs‘(𝑏𝑦))) = (𝑘 · (abs‘(𝑥𝑦))))
110106, 109breq12d 4631 . . . . . . . . . . 11 (𝑏 = 𝑥 → ((abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦))) ↔ (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦)))))
111103, 110imbi12d 334 . . . . . . . . . 10 (𝑏 = 𝑥 → ((𝑦 < 𝑏 → (abs‘((𝐹𝑏) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑏𝑦)))) ↔ (𝑦 < 𝑥 → (abs‘((𝐹𝑥) − (𝐹𝑦))) ≤ (𝑘 · (abs‘(𝑥𝑦))))))
112102, 111rspc2v 3310 . . . . . . . . 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 469 . . . . . . . 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 762 . . . . . . 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 477 . . . . . . . 8 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
116 fvres 6169 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹𝑦))
117116ad2antll 764 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹𝑦))
118 simpr 477 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
119 ffvelrn 6318 . . . . . . . . . . . . . . 15 (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
12061, 118, 119syl2an 494 . . . . . . . . . . . . . 14 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
121117, 120eqeltrrd 2699 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑦) ∈ ℝ)
122121recnd 10019 . . . . . . . . . . . 12 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹𝑦) ∈ ℂ)
12366, 122abssubd 14133 . . . . . . . . . . 11 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑥) − (𝐹𝑦))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
124123adantr 481 . . . . . . . . . 10 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘((𝐹𝑥) − (𝐹𝑦))) = (abs‘((𝐹𝑦) − (𝐹𝑥))))
12570ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
126125sseld 3586 . . . . . . . . . . . . . . 15 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ))
127125sseld 3586 . . . . . . . . . . . . . . 15 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ))
128126, 127anim12d 585 . . . . . . . . . . . . . 14 (((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
129128imp 445 . . . . . . . . . . . . 13 ((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
130 recn 9977 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
131 recn 9977 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
132 abssub 14007 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
133130, 131, 132syl2an 494 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘(𝑥𝑦)) = (abs‘(𝑦𝑥)))
136135oveq2d 6626 . . . . . . . . . 10 (((((𝜑𝐴𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (𝑘 · (abs‘(𝑥𝑦))) = (𝑘 · (abs‘(𝑦𝑥))))
137124, 136breq12d 4631 . . . . . . . . 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 10073 . . . . . . 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 1392 . . . . 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 2969 . . . 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 3012 . . 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 10096 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 384  w3o 1035   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  wss 3559  c0 3896   class class class wbr 4618  cres 5081  cima 5082  wf 5848  cfv 5852  (class class class)co 6610  pm cpm 7810  supcsup 8297  cc 9885  cr 9886  0cc0 9887   · cmul 9892  *cxr 10024   < clt 10025  cle 10026  cmin 10217  [,]cicc 12127  abscabs 13915  cnccncf 22598   D cdv 23546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965  ax-addf 9966  ax-mulf 9967
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7860  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-fi 8268  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-q 11740  df-rp 11784  df-xneg 11897  df-xadd 11898  df-xmul 11899  df-ioo 12128  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-starv 15884  df-sca 15885  df-vsca 15886  df-ip 15887  df-tset 15888  df-ple 15889  df-ds 15892  df-unif 15893  df-hom 15894  df-cco 15895  df-rest 16011  df-topn 16012  df-0g 16030  df-gsum 16031  df-topgen 16032  df-pt 16033  df-prds 16036  df-xrs 16090  df-qtop 16095  df-imas 16096  df-xps 16098  df-mre 16174  df-mrc 16175  df-acs 16177  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-submnd 17264  df-mulg 17469  df-cntz 17678  df-cmn 18123  df-psmet 19666  df-xmet 19667  df-met 19668  df-bl 19669  df-mopn 19670  df-fbas 19671  df-fg 19672  df-cnfld 19675  df-top 20627  df-topon 20644  df-topsp 20657  df-bases 20670  df-cld 20742  df-ntr 20743  df-cls 20744  df-nei 20821  df-lp 20859  df-perf 20860  df-cn 20950  df-cnp 20951  df-haus 21038  df-cmp 21109  df-tx 21284  df-hmeo 21477  df-fil 21569  df-fm 21661  df-flim 21662  df-flf 21663  df-xms 22044  df-ms 22045  df-tms 22046  df-cncf 22600  df-limc 23549  df-dv 23550
This theorem is referenced by:  c1lip2  23678
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