unblimceq0 - Mathbox for Asger C. Ipsen

	Mathbox for Asger C. Ipsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > unblimceq0		Structured version Visualization version GIF version

Theorem unblimceq0 36038

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	⊢ (𝜑 → 𝑆 ⊆ ℂ)
unblimceq0.1	⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
unblimceq0.2	⊢ (𝜑 → 𝐴 ∈ ℂ)
unblimceq0.3	⊢ (𝜑 → ∀𝑏 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))

**Assertion**
Ref	Expression
unblimceq0	⊢ (𝜑 → (𝐹 lim_ℂ 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.

Step	Hyp	Ref	Expression
1		1rp 13008	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
2	1	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 1 ∈ ℝ⁺)
3		breq2 5147	. . . . . . . . . . 11 ⊢ (𝑒 = 1 → ((abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒 ↔ (abs‘((𝐹‘𝑧) − 𝑦)) < 1))
4	3	imbi2d 339	. . . . . . . . . 10 ⊢ (𝑒 = 1 → (((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒) ↔ ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1)))
5	4	rexralbidv 3211	. . . . . . . . 9 ⊢ (𝑒 = 1 → (∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒) ↔ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1)))
6	5	notbid 317	. . . . . . . 8 ⊢ (𝑒 = 1 → (¬ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒) ↔ ¬ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1)))
7	6	adantl 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑒 = 1) → (¬ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒) ↔ ¬ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1)))
8		simprr1 1218	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → 𝑧 ≠ 𝐴)
9		simprr2 1219	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (abs‘(𝑧 − 𝐴)) < 𝑐)
10	8, 9	jca 510	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐))
11		1red 11243	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → 1 ∈ ℝ)
12		unblimceq0.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
13	12	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → 𝐹:𝑆⟶ℂ)
14	13	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → 𝐹:𝑆⟶ℂ)
15		simprl 769	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → 𝑧 ∈ 𝑆)
16	14, 15	ffvelcdmd 7089	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (𝐹‘𝑧) ∈ ℂ)
17		simplr 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
18	17	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → 𝑦 ∈ ℂ)
19	16, 18	subcld 11599	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → ((𝐹‘𝑧) − 𝑦) ∈ ℂ)
20	19	abscld 15413	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (abs‘((𝐹‘𝑧) − 𝑦)) ∈ ℝ)
21	16	abscld 15413	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (abs‘(𝐹‘𝑧)) ∈ ℝ)
22	17	abscld 15413	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → (abs‘𝑦) ∈ ℝ)
23	22	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (abs‘𝑦) ∈ ℝ)
24	21, 23	resubcld 11670	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → ((abs‘(𝐹‘𝑧)) − (abs‘𝑦)) ∈ ℝ)
25		1cnd 11237	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → 1 ∈ ℂ)
26	23	recnd 11270	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (abs‘𝑦) ∈ ℂ)
27	25, 26	pncand 11600	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → ((1 + (abs‘𝑦)) − (abs‘𝑦)) = 1)
28		1red 11243	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → 1 ∈ ℝ)
29	28, 22	readdcld 11271	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → (1 + (abs‘𝑦)) ∈ ℝ)
30	29	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (1 + (abs‘𝑦)) ∈ ℝ)
31		simprr3 1220	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧)))
32	30, 21, 23, 31	lesub1dd 11858	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → ((1 + (abs‘𝑦)) − (abs‘𝑦)) ≤ ((abs‘(𝐹‘𝑧)) − (abs‘𝑦)))
33	27, 32	eqbrtrrd 5167	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → 1 ≤ ((abs‘(𝐹‘𝑧)) − (abs‘𝑦)))
34	16, 18	abs2difd 15434	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → ((abs‘(𝐹‘𝑧)) − (abs‘𝑦)) ≤ (abs‘((𝐹‘𝑧) − 𝑦)))
35	11, 24, 20, 33, 34	letrd 11399	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → 1 ≤ (abs‘((𝐹‘𝑧) − 𝑦)))
36	11, 20, 35	lensymd 11393	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → ¬ (abs‘((𝐹‘𝑧) − 𝑦)) < 1)
37	10, 36	jcnd 163	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑧 ∈ 𝑆 ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))) → ¬ ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1))
38		breq2 5147	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑐 → ((abs‘(𝑧 − 𝐴)) < 𝑑 ↔ (abs‘(𝑧 − 𝐴)) < 𝑐))
39	38	3anbi2d 1437	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))) ↔ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧)))))
40	39	rexbidv 3169	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))) ↔ ∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧)))))
41		breq1 5146	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (1 + (abs‘𝑦)) → (𝑎 ≤ (abs‘(𝐹‘𝑧)) ↔ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))
42	41	3anbi3d 1438	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (1 + (abs‘𝑦)) → ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ 𝑎 ≤ (abs‘(𝐹‘𝑧))) ↔ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧)))))
43	42	rexbidv 3169	. . . . . . . . . . . . 13 ⊢ (𝑎 = (1 + (abs‘𝑦)) → (∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ 𝑎 ≤ (abs‘(𝐹‘𝑧))) ↔ ∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧)))))
44	43	ralbidv 3168	. . . . . . . . . . . 12 ⊢ (𝑎 = (1 + (abs‘𝑦)) → (∀𝑑 ∈ ℝ⁺ ∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ 𝑎 ≤ (abs‘(𝐹‘𝑧))) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧)))))
45		unblimceq0.0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
46		unblimceq0.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℂ)
47		unblimceq0.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑏 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))
48	45, 12, 46, 47	unblimceq0lem 36037	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑎 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ 𝑎 ≤ (abs‘(𝐹‘𝑧))))
49	48	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → ∀𝑎 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ 𝑎 ≤ (abs‘(𝐹‘𝑧))))
50		0lt1 11764	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → 0 < 1)
52	17	absge0d 15421	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → 0 ≤ (abs‘𝑦))
53	28, 22, 51, 52	addgtge0d 11816	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → 0 < (1 + (abs‘𝑦)))
54	29, 53	elrpd 13043	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → (1 + (abs‘𝑦)) ∈ ℝ⁺)
55	44, 49, 54	rspcdva 3603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → ∀𝑑 ∈ ℝ⁺ ∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑑 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))
56		simpr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → 𝑐 ∈ ℝ⁺)
57	40, 55, 56	rspcdva 3603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → ∃𝑧 ∈ 𝑆 (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐 ∧ (1 + (abs‘𝑦)) ≤ (abs‘(𝐹‘𝑧))))
58	37, 57	reximddv 3161	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → ∃𝑧 ∈ 𝑆 ¬ ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1))
59		rexnal 3090	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑆 ¬ ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1) ↔ ¬ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1))
60	58, 59	sylib 217	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑐 ∈ ℝ⁺) → ¬ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1))
61	60	nrexdv 3139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ¬ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 1))
62	2, 7, 61	rspcedvd 3604	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ∃𝑒 ∈ ℝ⁺ ¬ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒))
63		rexnal 3090	. . . . . 6 ⊢ (∃𝑒 ∈ ℝ⁺ ¬ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒) ↔ ¬ ∀𝑒 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒))
64	62, 63	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ¬ ∀𝑒 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒))
65	64	ex 411	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℂ → ¬ ∀𝑒 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))
66		imnan 398	. . . 4 ⊢ ((𝑦 ∈ ℂ → ¬ ∀𝑒 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)) ↔ ¬ (𝑦 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))
67	65, 66	sylib 217	. . 3 ⊢ (𝜑 → ¬ (𝑦 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))
68	12, 45, 46	ellimc3 25824	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹 lim_ℂ 𝐴) ↔ (𝑦 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑐 ∈ ℝ⁺ ∀𝑧 ∈ 𝑆 ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑐) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒))))
69	67, 68	mtbird 324	. 2 ⊢ (𝜑 → ¬ 𝑦 ∈ (𝐹 lim_ℂ 𝐴))
70	69	eq0rdv 4400	1 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐴) = ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ⊆ wss 3940 ∅c0 4318 class class class wbr 5143 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 ℂcc 11134 ℝcr 11135 0cc0 11136 1c1 11137 + caddc 11139 < clt 11276 ≤ cle 11277 − cmin 11472 ℝ⁺crp 13004 abscabs 15211 lim_ℂ climc 25807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fi 9432 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-fz 13515 df-seq 13997 df-exp 14057 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-struct 17113 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-mulr 17244 df-starv 17245 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-rest 17401 df-topn 17402 df-topgen 17422 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cnp 23148 df-xms 24242 df-ms 24243 df-limc 25811

This theorem is referenced by: unbdqndv1 36039

W3C validator