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Theorem unblimceq0 32834
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 12038 . . . . . . . . 9 1 ∈ ℝ+
21a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ ℂ) → 1 ∈ ℝ+)
3 breq2 4791 . . . . . . . . . . . . 13 (𝑒 = 1 → ((abs‘((𝐹𝑧) − 𝑦)) < 𝑒 ↔ (abs‘((𝐹𝑧) − 𝑦)) < 1))
43imbi2d 329 . . . . . . . . . . . 12 (𝑒 = 1 → (((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1)))
54ralbidv 3135 . . . . . . . . . . 11 (𝑒 = 1 → (∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1)))
65rexbidv 3200 . . . . . . . . . 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 307 . . . . . . . . 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 467 . . . . . . . 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 1272 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑧𝐴)
10 simprr2 1274 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘(𝑧𝐴)) < 𝑐)
119, 10jca 501 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐))
12 1red 10260 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 1 ∈ ℝ)
1312adantr 466 . . . . . . . . . . . . . . . 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 705 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
1615adantr 466 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝐹:𝑆⟶ℂ)
17 simprl 754 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑧𝑆)
1816, 17ffvelrnd 6505 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (𝐹𝑧) ∈ ℂ)
1918abscld 14382 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘(𝐹𝑧)) ∈ ℝ)
20 simplr 752 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝑦 ∈ ℂ)
2120abscld 14382 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (abs‘𝑦) ∈ ℝ)
2221adantr 466 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘𝑦) ∈ ℝ)
2319, 22resubcld 10663 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((abs‘(𝐹𝑧)) − (abs‘𝑦)) ∈ ℝ)
2420adantr 466 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑦 ∈ ℂ)
2518, 24subcld 10597 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((𝐹𝑧) − 𝑦) ∈ ℂ)
2625abscld 14382 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘((𝐹𝑧) − 𝑦)) ∈ ℝ)
27 1cnd 10261 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ∈ ℂ)
2822recnd 10273 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘𝑦) ∈ ℂ)
2927, 28pncand 10598 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((1 + (abs‘𝑦)) − (abs‘𝑦)) = 1)
3029eqcomd 2777 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 = ((1 + (abs‘𝑦)) − (abs‘𝑦)))
31 simprr3 1276 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))
3212, 21readdcld 10274 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (1 + (abs‘𝑦)) ∈ ℝ)
3332adantr 466 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (1 + (abs‘𝑦)) ∈ ℝ)
3433, 19, 22lesub1d 10839 . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . 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 4809 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ≤ ((abs‘(𝐹𝑧)) − (abs‘𝑦)))
3718, 24abs2difd 14403 . . . . . . . . . . . . . . . 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 10399 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ≤ (abs‘((𝐹𝑧) − 𝑦)))
3913, 26lenltd 10388 . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ¬ (abs‘((𝐹𝑧) − 𝑦)) < 1)
4111, 40jca 501 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) ∧ ¬ (abs‘((𝐹𝑧) − 𝑦)) < 1))
42 pm4.61 391 . . . . . . . . . . . . 13 (¬ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1) ↔ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) ∧ ¬ (abs‘((𝐹𝑧) − 𝑦)) < 1))
4341, 42sylibr 224 . . . . . . . . . . . 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 4791 . . . . . . . . . . . . . . 15 (𝑑 = 𝑐 → ((abs‘(𝑧𝐴)) < 𝑑 ↔ (abs‘(𝑧𝐴)) < 𝑐))
45443anbi2d 1552 . . . . . . . . . . . . . 14 (𝑑 = 𝑐 → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))) ↔ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
4645rexbidv 3200 . . . . . . . . . . . . 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 4790 . . . . . . . . . . . . . . . . 17 (𝑎 = (1 + (abs‘𝑦)) → (𝑎 ≤ (abs‘(𝐹𝑧)) ↔ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
48473anbi3d 1553 . . . . . . . . . . . . . . . 16 (𝑎 = (1 + (abs‘𝑦)) → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))) ↔ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
4948rexbidv 3200 . . . . . . . . . . . . . . 15 (𝑎 = (1 + (abs‘𝑦)) → (∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))) ↔ ∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
5049ralbidv 3135 . . . . . . . . . . . . . 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 32833 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑎 ∈ ℝ+𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))))
5554ad2antrr 705 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∀𝑎 ∈ ℝ+𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))))
56 0lt1 10755 . . . . . . . . . . . . . . . . 17 0 < 1
5756a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 < 1)
5820absge0d 14390 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 ≤ (abs‘𝑦))
5912, 21, 57, 58addgtge0d 32832 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 < (1 + (abs‘𝑦)))
6032, 59elrpd 12071 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (1 + (abs‘𝑦)) ∈ ℝ+)
6150, 55, 60rspcdva 3466 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∀𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
62 simpr 471 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝑐 ∈ ℝ+)
6346, 61, 62rspcdva 3466 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
6443, 63reximddv 3166 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∃𝑧𝑆 ¬ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
65 rexnal 3143 . . . . . . . . . . 11 (∃𝑧𝑆 ¬ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1) ↔ ¬ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
6664, 65sylib 208 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ¬ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
6766ralrimiva 3115 . . . . . . . . 9 ((𝜑𝑦 ∈ ℂ) → ∀𝑐 ∈ ℝ+ ¬ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
68 ralnex 3141 . . . . . . . . 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 208 . . . . . . . 8 ((𝜑𝑦 ∈ ℂ) → ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
702, 8, 69rspcedvd 3467 . . . . . . 7 ((𝜑𝑦 ∈ ℂ) → ∃𝑒 ∈ ℝ+ ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))
71 rexnal 3143 . . . . . . 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 208 . . . . . 6 ((𝜑𝑦 ∈ ℂ) → ¬ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))
7372ex 397 . . . . 5 (𝜑 → (𝑦 ∈ ℂ → ¬ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
74 imnan 386 . . . . 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 208 . . . 4 (𝜑 → ¬ (𝑦 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
7614, 51, 52ellimc3 23862 . . . 4 (𝜑 → (𝑦 ∈ (𝐹 lim 𝐴) ↔ (𝑦 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))))
7775, 76mtbird 314 . . 3 (𝜑 → ¬ 𝑦 ∈ (𝐹 lim 𝐴))
7877alrimiv 2007 . 2 (𝜑 → ∀𝑦 ¬ 𝑦 ∈ (𝐹 lim 𝐴))
79 eq0 4077 . 2 ((𝐹 lim 𝐴) = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ (𝐹 lim 𝐴))
8078, 79sylibr 224 1 (𝜑 → (𝐹 lim 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  wss 3723  c0 4063   class class class wbr 4787  wf 6026  cfv 6030  (class class class)co 6795  cc 10139  cr 10140  0cc0 10141  1c1 10142   + caddc 10144   < clt 10279  cle 10280  cmin 10471  +crp 12034  abscabs 14181   lim climc 23845
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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-pre-sup 10219
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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-oadd 7720  df-er 7899  df-map 8014  df-pm 8015  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-fi 8476  df-sup 8507  df-inf 8508  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-div 10890  df-nn 11226  df-2 11284  df-3 11285  df-4 11286  df-5 11287  df-6 11288  df-7 11289  df-8 11290  df-9 11291  df-n0 11499  df-z 11584  df-dec 11700  df-uz 11893  df-q 11996  df-rp 12035  df-xneg 12150  df-xadd 12151  df-xmul 12152  df-fz 12533  df-seq 13008  df-exp 13067  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-plusg 16161  df-mulr 16162  df-starv 16163  df-tset 16167  df-ple 16168  df-ds 16171  df-unif 16172  df-rest 16290  df-topn 16291  df-topgen 16311  df-psmet 19952  df-xmet 19953  df-met 19954  df-bl 19955  df-mopn 19956  df-cnfld 19961  df-top 20918  df-topon 20935  df-topsp 20957  df-bases 20970  df-cnp 21252  df-xms 22344  df-ms 22345  df-limc 23849
This theorem is referenced by:  unbdqndv1  32835
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