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Theorem unblimceq0 32623
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 11874 . . . . . . . . 9 1 ∈ ℝ+
21a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ ℂ) → 1 ∈ ℝ+)
3 breq2 4689 . . . . . . . . . . . . 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 3015 . . . . . . . . . . 11 (𝑒 = 1 → (∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1)))
65rexbidv 3081 . . . . . . . . . 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 481 . . . . . . . 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 1129 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑧𝐴)
10 simprr2 1130 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘(𝑧𝐴)) < 𝑐)
119, 10jca 553 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐))
12 1red 10093 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 1 ∈ ℝ)
1312adantr 480 . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
1615adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝐹:𝑆⟶ℂ)
17 simprl 809 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑧𝑆)
1816, 17ffvelrnd 6400 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (𝐹𝑧) ∈ ℂ)
1918abscld 14219 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘(𝐹𝑧)) ∈ ℝ)
20 simplr 807 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝑦 ∈ ℂ)
2120abscld 14219 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (abs‘𝑦) ∈ ℝ)
2221adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘𝑦) ∈ ℝ)
2319, 22resubcld 10496 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((abs‘(𝐹𝑧)) − (abs‘𝑦)) ∈ ℝ)
2420adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 𝑦 ∈ ℂ)
2518, 24subcld 10430 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((𝐹𝑧) − 𝑦) ∈ ℂ)
2625abscld 14219 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘((𝐹𝑧) − 𝑦)) ∈ ℝ)
27 1cnd 10094 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ∈ ℂ)
2822recnd 10106 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (abs‘𝑦) ∈ ℂ)
2927, 28pncand 10431 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((1 + (abs‘𝑦)) − (abs‘𝑦)) = 1)
3029eqcomd 2657 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 = ((1 + (abs‘𝑦)) − (abs‘𝑦)))
31 simprr3 1131 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))
3212, 21readdcld 10107 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (1 + (abs‘𝑦)) ∈ ℝ)
3332adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → (1 + (abs‘𝑦)) ∈ ℝ)
3433, 19, 22lesub1d 10672 . . . . . . . . . . . . . . . . . 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 4707 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ≤ ((abs‘(𝐹𝑧)) − (abs‘𝑦)))
3718, 24abs2difd 14240 . . . . . . . . . . . . . . . 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 10232 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → 1 ≤ (abs‘((𝐹𝑧) − 𝑦)))
3913, 26lenltd 10221 . . . . . . . . . . . . . . 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 553 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) ∧ (𝑧𝑆 ∧ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))) → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) ∧ ¬ (abs‘((𝐹𝑧) − 𝑦)) < 1))
42 pm4.61 441 . . . . . . . . . . . . 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 4689 . . . . . . . . . . . . . . 15 (𝑑 = 𝑐 → ((abs‘(𝑧𝐴)) < 𝑑 ↔ (abs‘(𝑧𝐴)) < 𝑐))
45443anbi2d 1444 . . . . . . . . . . . . . 14 (𝑑 = 𝑐 → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))) ↔ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
4645rexbidv 3081 . . . . . . . . . . . . 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 4688 . . . . . . . . . . . . . . . . 17 (𝑎 = (1 + (abs‘𝑦)) → (𝑎 ≤ (abs‘(𝐹𝑧)) ↔ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
48473anbi3d 1445 . . . . . . . . . . . . . . . 16 (𝑎 = (1 + (abs‘𝑦)) → ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))) ↔ (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
4948rexbidv 3081 . . . . . . . . . . . . . . 15 (𝑎 = (1 + (abs‘𝑦)) → (∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))) ↔ ∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧)))))
5049ralbidv 3015 . . . . . . . . . . . . . 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 32622 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑎 ∈ ℝ+𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))))
5554ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∀𝑎 ∈ ℝ+𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑𝑎 ≤ (abs‘(𝐹𝑧))))
56 0lt1 10588 . . . . . . . . . . . . . . . . 17 0 < 1
5756a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 < 1)
5820absge0d 14227 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 ≤ (abs‘𝑦))
5912, 21, 57, 58addgtge0d 32621 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 0 < (1 + (abs‘𝑦)))
6032, 59elrpd 11907 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → (1 + (abs‘𝑦)) ∈ ℝ+)
6150, 55, 60rspcdva 3347 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∀𝑑 ∈ ℝ+𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
62 simpr 476 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → 𝑐 ∈ ℝ+)
6346, 61, 62rspcdva 3347 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∃𝑧𝑆 (𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹𝑧))))
6443, 63reximddv 3047 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ+) → ∃𝑧𝑆 ¬ ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
65 rexnal 3024 . . . . . . . . . . 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 2995 . . . . . . . . 9 ((𝜑𝑦 ∈ ℂ) → ∀𝑐 ∈ ℝ+ ¬ ∀𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 1))
68 ralnex 3021 . . . . . . . . 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 3348 . . . . . . 7 ((𝜑𝑦 ∈ ℂ) → ∃𝑒 ∈ ℝ+ ¬ ∃𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))
71 rexnal 3024 . . . . . . 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 449 . . . . 5 (𝜑 → (𝑦 ∈ ℂ → ¬ ∀𝑒 ∈ ℝ+𝑐 ∈ ℝ+𝑧𝑆 ((𝑧𝐴 ∧ (abs‘(𝑧𝐴)) < 𝑐) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
74 imnan 437 . . . . 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 23688 . . . 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 1895 . 2 (𝜑 → ∀𝑦 ¬ 𝑦 ∈ (𝐹 lim 𝐴))
79 eq0 3962 . 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 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  +crp 11870  abscabs 14018   lim climc 23671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-fz 12365  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-rest 16130  df-topn 16131  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cnp 21080  df-xms 22172  df-ms 22173  df-limc 23675
This theorem is referenced by:  unbdqndv1  32624
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