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Theorem ellimc3apf 14065

Description: Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)

**Hypotheses**
Ref	Expression
ellimc3.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
ellimc3.a	⊢ (𝜑 → 𝐴 ⊆ ℂ)
ellimc3.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
ellimc3.nf	⊢ Ⅎ𝑧𝐹

**Assertion**
Ref	Expression
ellimc3apf	⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))

Distinct variable groups: 𝑧,𝐴 𝑥,𝐵,𝑦,𝑧 𝑥,𝐶,𝑦,𝑧 𝑥,𝐹,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑥,𝑦) 𝐹(𝑧)

Proof of Theorem ellimc3apf Dummy variables 𝑓 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 7934	. . . . . . 7 ⊢ ℂ ∈ V
2	1	a1i 9	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
3		ellimc3.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
4		ellimc3.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
5		elpm2r 6665	. . . . . 6 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
6	2, 2, 3, 4, 5	syl22anc 1239	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℂ))
7		ellimc3.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8	1	rabex 4147	. . . . . 6 ⊢ {𝑢 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))} ∈ V
9	8	a1i 9	. . . . 5 ⊢ (𝜑 → {𝑢 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))} ∈ V)
10		simpl 109	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → 𝑓 = 𝐹)
11	10	dmeqd 4829	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → dom 𝑓 = dom 𝐹)
12	10, 11	feq12d 5355	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
13	11	sseq1d 3184	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (dom 𝑓 ⊆ ℂ ↔ dom 𝐹 ⊆ ℂ))
14	12, 13	anbi12d 473	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ)))
15		simpr 110	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → 𝑤 = 𝐵)
16	15	eleq1d 2246	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (𝑤 ∈ ℂ ↔ 𝐵 ∈ ℂ))
17		nfcv 2319	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧dom 𝑓
18		ellimc3.nf	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧𝐹
19	18	nfdm 4871	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧dom 𝐹
20	17, 19	raleqf 2668	. . . . . . . . . . . . 13 ⊢ (dom 𝑓 = dom 𝐹 → (∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥)))
21	11, 20	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥)))
22	18	nfeq2 2331	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 𝑓 = 𝐹
23		nfv 1528	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 𝑤 = 𝐵
24	22, 23	nfan 1565	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧(𝑓 = 𝐹 ∧ 𝑤 = 𝐵)
25	15	breq2d 4015	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (𝑧 # 𝑤 ↔ 𝑧 # 𝐵))
26	15	oveq2d 5890	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (𝑧 − 𝑤) = (𝑧 − 𝐵))
27	26	fveq2d 5519	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (abs‘(𝑧 − 𝑤)) = (abs‘(𝑧 − 𝐵)))
28	27	breq1d 4013	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → ((abs‘(𝑧 − 𝑤)) < 𝑦 ↔ (abs‘(𝑧 − 𝐵)) < 𝑦))
29	25, 28	anbi12d 473	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → ((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) ↔ (𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦)))
30	10	fveq1d 5517	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (𝑓‘𝑧) = (𝐹‘𝑧))
31	30	fvoveq1d 5896	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (abs‘((𝑓‘𝑧) − 𝑢)) = (abs‘((𝐹‘𝑧) − 𝑢)))
32	31	breq1d 4013	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → ((abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥 ↔ (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥))
33	29, 32	imbi12d 234	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥) ↔ ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))
34	24, 33	ralbid 2475	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (∀𝑧 ∈ dom 𝐹((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))
35	21, 34	bitrd 188	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))
36	35	rexbidv 2478	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥) ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))
37	36	ralbidv 2477	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))
38	16, 37	anbi12d 473	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → ((𝑤 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥)) ↔ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥))))
39	14, 38	anbi12d 473	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → (((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑤 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥))) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))))
40	39	rabbidv 2726	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐵) → {𝑢 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑤 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥)))} = {𝑢 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))})
41		df-limced 14061	. . . . . 6 ⊢ lim_ℂ = (𝑓 ∈ (ℂ ↑_pm ℂ), 𝑤 ∈ ℂ ↦ {𝑢 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑤 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑤 ∧ (abs‘(𝑧 − 𝑤)) < 𝑦) → (abs‘((𝑓‘𝑧) − 𝑢)) < 𝑥)))})
42	40, 41	ovmpoga 6003	. . . . 5 ⊢ ((𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝐵 ∈ ℂ ∧ {𝑢 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))} ∈ V) → (𝐹 lim_ℂ 𝐵) = {𝑢 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))})
43	6, 7, 9, 42	syl3anc 1238	. . . 4 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) = {𝑢 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))})
44	43	eleq2d 2247	. . 3 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ 𝐶 ∈ {𝑢 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))}))
45		oveq2 5882	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐶 → ((𝐹‘𝑧) − 𝑢) = ((𝐹‘𝑧) − 𝐶))
46	45	fveq2d 5519	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐶 → (abs‘((𝐹‘𝑧) − 𝑢)) = (abs‘((𝐹‘𝑧) − 𝐶)))
47	46	breq1d 4013	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → ((abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥 ↔ (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))
48	47	imbi2d 230	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → (((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥) ↔ ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
49	48	ralbidv 2477	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
50	49	rexbidv 2478	. . . . . . 7 ⊢ (𝑢 = 𝐶 → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥) ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
51	50	ralbidv 2477	. . . . . 6 ⊢ (𝑢 = 𝐶 → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
52	51	anbi2d 464	. . . . 5 ⊢ (𝑢 = 𝐶 → ((𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)) ↔ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
53	52	anbi2d 464	. . . 4 ⊢ (𝑢 = 𝐶 → (((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥))) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))
54	53	elrab 2893	. . 3 ⊢ (𝐶 ∈ {𝑢 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝑢)) < 𝑥)))} ↔ (𝐶 ∈ ℂ ∧ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))
55	44, 54	bitrdi 196	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))))
56	7	biantrurd 305	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
57	3	fdmd 5372	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴)
58	57	feq2d 5353	. . . . . 6 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:𝐴⟶ℂ))
59	3, 58	mpbird 167	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
60	57, 4	eqsstrd 3191	. . . . 5 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
61		ibar 301	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) → ((𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))
62	59, 60, 61	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))
63	56, 62	bitrd 188	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))
64	63	anbi2d 464	. 2 ⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝐶 ∈ ℂ ∧ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))))
65		nfcv 2319	. . . . . . 7 ⊢ Ⅎ𝑧𝐴
66	19, 65	raleqf 2668	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → (∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
67	57, 66	syl 14	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
68	67	rexbidv 2478	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
69	68	ralbidv 2477	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
70	69	anbi2d 464	. 2 ⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
71	55, 64, 70	3bitr2d 216	1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Ⅎwnfc 2306 ∀wral 2455 ∃wrex 2456 {crab 2459 Vcvv 2737 ⊆ wss 3129 class class class wbr 4003 dom cdm 4626 ⟶wf 5212 ‘cfv 5216 (class class class)co 5874 ↑_pm cpm 6648 ℂcc 7808 < clt 7991 − cmin 8127 # cap 8537 ℝ⁺crp 9652 abscabs 11005 lim_ℂ climc 14059

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pm 6650 df-limced 14061

This theorem is referenced by: ellimc3ap 14066 limcmpted 14068

W3C validator