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Theorem unblimceq0 32940
Description: If 𝐹 is unbounded near 𝐴 it has no limit at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
Hypotheses
Ref Expression
unblimceq0.0 (𝜑𝑆 ⊆ ℂ)
unblimceq0.1 (𝜑𝐹:𝑆⟶ℂ)
unblimceq0.2 (𝜑𝐴 ∈ ℂ)
unblimceq0.3 (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝑆 ((abs‘(𝑥𝐴)) < 𝑑𝑏 ≤ (abs‘(𝐹𝑥))))
Assertion
Ref Expression
unblimceq0 (𝜑 → (𝐹 lim 𝐴) = ∅)
Distinct variable groups:   𝐴,𝑏,𝑑,𝑥   𝐹,𝑏,𝑑,𝑥   𝑆,𝑏,𝑑,𝑥   𝜑,𝑏,𝑑,𝑥

Proof of Theorem unblimceq0
Dummy variables 𝑎 𝑐 𝑦 𝑧 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1rp 12035 . . . . . . . . 9 1 ∈ ℝ+
21a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ ℂ) → 1 ∈ ℝ+)
3 breq2 4815 . . . . . . . . . . . . 13 (𝑒 = 1 → ((abs‘((𝐹𝑧) − 𝑦)) < 𝑒 ↔ (abs‘((𝐹𝑧) − 𝑦)) < 1))
43imbi2d 331 . . . . . . . . . . . 12 (𝑒 = 1 → (((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1)))
54ralbidv 3133 . . . . . . . . . . 11 (𝑒 = 1 → (∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1)))
65rexbidv 3199 . . . . . . . . . 10 (𝑒 = 1 → (∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1)))
76notbid 309 . . . . . . . . 9 (𝑒 = 1 → (¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1)))
87adantl 473 . . . . . . . 8 (((𝜑𝑦 ∈ ℂ) ∧ 𝑒 = 1) → (¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1)))
9 simprr1 1287 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑧𝐴)
10 simprr2 1289 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘(𝑧𝐴)) < 𝑐)
119, 10jca 507 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐))
12 1red 10296 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 1 ∈ ℝ)
1312adantr 472 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ∈ ℝ)
14 unblimceq0.1 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐹:𝑆⟶ℂ)
1514ad2antrr 717 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
1615adantr 472 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝐹:𝑆⟶ℂ)
17 simprl 787 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑧𝑆)
1816, 17ffvelrnd 6552 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (𝐹𝑧) ∈ ℂ)
1918abscld 14463 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘(𝐹𝑧)) ∈ ℝ)
20 simplr 785 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝑦 ∈ ℂ)
2120abscld 14463 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (abs‘𝑦) ∈ ℝ)
2221adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘𝑦) ∈ ℝ)
2319, 22resubcld 10714 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((abs‘(𝐹𝑧)) − (abs‘𝑦)) ∈ ℝ)
2420adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑦 ∈ ℂ)
2518, 24subcld 10648 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((𝐹𝑧) − 𝑦) ∈ ℂ)
2625abscld 14463 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘((𝐹𝑧) − 𝑦)) ∈ ℝ)
27 1cnd 10290 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ∈ ℂ)
2822recnd 10324 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘𝑦) ∈ ℂ)
2927, 28pncand 10649 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((1 + (abs‘𝑦)) − (abs‘𝑦)) = 1)
3029eqcomd 2771 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 = ((1 + (abs‘𝑦)) − (abs‘𝑦)))
31 simprr3 1291 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))
3212, 21readdcld 10325 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (1 + (abs‘𝑦)) ∈ ℝ)
3332adantr 472 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (1 + (abs‘𝑦)) ∈ ℝ)
3433, 19, 22lesub1d 10890 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)) ↔ ((1 + (abs‘𝑦)) − (abs‘𝑦)) ≤ ((abs‘(𝐹𝑧)) − (abs‘𝑦))))
3531, 34mpbid 223 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((1 + (abs‘𝑦)) − (abs‘𝑦)) ≤ ((abs‘(𝐹𝑧)) − (abs‘𝑦)))
3630, 35eqbrtrd 4833 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ≤ ((abs‘(𝐹𝑧)) − (abs‘𝑦)))
3718, 24abs2difd 14484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((abs‘(𝐹𝑧)) − (abs‘𝑦)) ≤ (abs‘((𝐹𝑧) − 𝑦)))
3813, 23, 26, 36, 37letrd 10450 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ≤ (abs‘((𝐹𝑧) − 𝑦)))
3913, 26lenltd 10439 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (1 ≤ (abs‘((𝐹𝑧) − 𝑦)) ↔ ¬ (abs‘((𝐹𝑧) − 𝑦)) < 1))
4038, 39mpbid 223 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ¬ (abs‘((𝐹𝑧) − 𝑦)) < 1)
4111, 40jca 507 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) ∧ ¬ (abs‘((𝐹𝑧) − 𝑦)) < 1))
42 pm4.61 393 . . . . . . . . . . . . 13 (¬ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1) ↔ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) ∧ ¬ (abs‘((𝐹𝑧) − 𝑦)) < 1))
4341, 42sylibr 225 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ¬ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
44 breq2 4815 . . . . . . . . . . . . . . 15 (𝑑 = 𝑐 → ((abs‘(𝑧𝐴)) < 𝑑 ↔ (abs‘(𝑧𝐴)) < 𝑐))
45443anbi2d 1565 . . . . . . . . . . . . . 14 (𝑑 = 𝑐 → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))) ↔ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
4645rexbidv 3199 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → (∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))) ↔ ∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
47 breq1 4814 . . . . . . . . . . . . . . . . 17 (𝑎 = (1 + (abs‘𝑦)) → (𝑎 ≤ (abs‘(𝐹𝑧)) ↔ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
48473anbi3d 1566 . . . . . . . . . . . . . . . 16 (𝑎 = (1 + (abs‘𝑦)) → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))) ↔ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
4948rexbidv 3199 . . . . . . . . . . . . . . 15 (𝑎 = (1 + (abs‘𝑦)) → (∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))) ↔ ∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
5049ralbidv 3133 . . . . . . . . . . . . . 14 (𝑎 = (1 + (abs‘𝑦)) → (∀𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))) ↔ ∀𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
51 unblimceq0.0 . . . . . . . . . . . . . . . 16 (𝜑𝑆 ⊆ ℂ)
52 unblimceq0.2 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ ℂ)
53 unblimceq0.3 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝑆 ((abs‘(𝑥𝐴)) < 𝑑𝑏 ≤ (abs‘(𝐹𝑥))))
5451, 14, 52, 53unblimceq0lem 32939 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑎 ∈ ℝ+𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))))
5554ad2antrr 717 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∀𝑎 ∈ ℝ+𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))))
56 0lt1 10806 . . . . . . . . . . . . . . . . 17 0 < 1
5756a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 < 1)
5820absge0d 14471 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 ≤ (abs‘𝑦))
5912, 21, 57, 58addgtge0d 32938 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 < (1 + (abs‘𝑦)))
6032, 59elrpd 12070 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (1 + (abs‘𝑦)) ∈ ℝ+)
6150, 55, 60rspcdva 3468 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∀𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
62 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝑐 ∈ ℝ+)
6346, 61, 62rspcdva 3468 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
6443, 63reximddv 3164 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∃𝑧𝑆 ¬ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
65 rexnal 3141 . . . . . . . . . . 11 (∃𝑧𝑆 ¬ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1) ↔ ¬ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
6664, 65sylib 209 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ¬ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
6766ralrimiva 3113 . . . . . . . . 9 ((𝜑𝑦 ∈ ℂ) → ∀𝑐 ∈ ℝ+ ¬ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
68 ralnex 3139 . . . . . . . . 9 (∀𝑐 ∈ ℝ+ ¬ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1) ↔ ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
6967, 68sylib 209 . . . . . . . 8 ((𝜑𝑦 ∈ ℂ) → ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
702, 8, 69rspcedvd 3469 . . . . . . 7 ((𝜑𝑦 ∈ ℂ) → ∃𝑒 ∈ ℝ+ ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))
71 rexnal 3141 . . . . . . 7 (∃𝑒 ∈ ℝ+ ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ¬ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))
7270, 71sylib 209 . . . . . 6 ((𝜑𝑦 ∈ ℂ) → ¬ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))
7372ex 401 . . . . 5 (𝜑 → (𝑦 ∈ ℂ → ¬ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
74 imnan 388 . . . . 5 ((𝑦 ∈ ℂ → ¬ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)) ↔ ¬ (𝑦 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
7573, 74sylib 209 . . . 4 (𝜑 → ¬ (𝑦 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
7614, 51, 52ellimc3 23937 . . . 4 (𝜑 → (𝑦 ∈ (𝐹 lim 𝐴) ↔ (𝑦 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))))
7775, 76mtbird 316 . . 3 (𝜑 → ¬ 𝑦 ∈ (𝐹 lim 𝐴))
7877alrimiv 2022 . 2 (𝜑 → ∀𝑦 ¬ 𝑦 ∈ (𝐹 lim 𝐴))
79 eq0 4095 . 2 ((𝐹 lim 𝐴) = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ (𝐹 lim 𝐴))
8078, 79sylibr 225 1 (𝜑 → (𝐹 lim 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  wss 3734  c0 4081   class class class wbr 4811  wf 6066  cfv 6070  (class class class)co 6844  cc 10189  cr 10190  0cc0 10191  1c1 10192   + caddc 10194   < clt 10330  cle 10331  cmin 10522  +crp 12031  abscabs 14262   lim climc 23920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-pm 8065  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-fi 8526  df-sup 8557  df-inf 8558  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-4 11339  df-5 11340  df-6 11341  df-7 11342  df-8 11343  df-9 11344  df-n0 11541  df-z 11627  df-dec 11744  df-uz 11890  df-q 11993  df-rp 12032  df-xneg 12149  df-xadd 12150  df-xmul 12151  df-fz 12537  df-seq 13012  df-exp 13071  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-struct 16135  df-ndx 16136  df-slot 16137  df-base 16139  df-plusg 16230  df-mulr 16231  df-starv 16232  df-tset 16236  df-ple 16237  df-ds 16239  df-unif 16240  df-rest 16352  df-topn 16353  df-topgen 16373  df-psmet 20014  df-xmet 20015  df-met 20016  df-bl 20017  df-mopn 20018  df-cnfld 20023  df-top 20981  df-topon 20998  df-topsp 21020  df-bases 21033  df-cnp 21315  df-xms 22407  df-ms 22408  df-limc 23924
This theorem is referenced by:  unbdqndv1  32941
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