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Theorem ellimc3 25780

Description: Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.)

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

**Assertion**
Ref	Expression
ellimc3	⊢ (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: 𝑥,𝑦,𝑧,𝐴 𝑥,𝐵,𝑦,𝑧 𝑥,𝐶,𝑦,𝑧 𝜑,𝑥,𝑦,𝑧 𝑥,𝐹,𝑦,𝑧

Proof of Theorem ellimc3 Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellimc3.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
2		ellimc3.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
3		ellimc3.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4		eqid 2729	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	1, 2, 3, 4	ellimc2 25778	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
6		cnxmet 24660	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
7		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
8		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
9		blcntr 24301	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
10	6, 7, 8, 9	mp3an2i 1468	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
11		rpxr 12961	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ^*)
12	11	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ^*)
13	4	cnfldtopn 24669	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blopn 24388	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ^*) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂ_fld))
15	6, 7, 12, 14	mp3an2i 1468	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂ_fld))
16		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
17		sseq2 3973	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
18	17	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
19	18	rexbidv 3157	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
20	16, 19	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) ↔ (𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))))
21	20	rspcv 3584	. . . . . . . . 9 ⊢ ((𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂ_fld) → (∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → (𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))))
22	15, 21	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → (∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → (𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))))
23	10, 22	mpid 44	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → (∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
24	13	mopni2 24381	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑣 ∈ (TopOpen‘ℂ_fld) ∧ 𝐵 ∈ 𝑣) → ∃𝑦 ∈ ℝ⁺ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
25	6, 24	mp3an1 1450	. . . . . . . . . 10 ⊢ ((𝑣 ∈ (TopOpen‘ℂ_fld) ∧ 𝐵 ∈ 𝑣) → ∃𝑦 ∈ ℝ⁺ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
26		ssrin 4205	. . . . . . . . . . . . 13 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
27		imass2 6073	. . . . . . . . . . . . 13 ⊢ (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
28		sstr2 3953	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
29	26, 27, 28	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
30	29	com12 32	. . . . . . . . . . 11 ⊢ ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
31	30	reximdv 3148	. . . . . . . . . 10 ⊢ ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → (∃𝑦 ∈ ℝ⁺ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
32	25, 31	syl5com 31	. . . . . . . . 9 ⊢ ((𝑣 ∈ (TopOpen‘ℂ_fld) ∧ 𝐵 ∈ 𝑣) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
33	32	impr 454	. . . . . . . 8 ⊢ ((𝑣 ∈ (TopOpen‘ℂ_fld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))) → ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))
34	33	rexlimiva 3126	. . . . . . 7 ⊢ (∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))
35	23, 34	syl6 35	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → (∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
36	35	ralrimdva 3133	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
37	13	mopni2 24381	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝐶 ∈ 𝑢) → ∃𝑥 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
38	6, 37	mp3an1 1450	. . . . . . . . 9 ⊢ ((𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝐶 ∈ 𝑢) → ∃𝑥 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
39		r19.29r 3096	. . . . . . . . . . 11 ⊢ ((∃𝑥 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑥 ∈ ℝ⁺ ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 ∧ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
40	3	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
41		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
42	41	rpxrd 12996	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
43	13	blopn 24388	. . . . . . . . . . . . . . . . 17 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ^*) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂ_fld))
44	6, 40, 42, 43	mp3an2i 1468	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂ_fld))
45		blcntr 24301	. . . . . . . . . . . . . . . . 17 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ⁺) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
46	6, 40, 41, 45	mp3an2i 1468	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
47		eleq2 2817	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
48		ineq1 4176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
49	48	imaeq2d 6031	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
50	49	sseq1d 3978	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
51	47, 50	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → ((𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ (𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
52	51	rspcev 3588	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂ_fld) ∧ (𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
53	52	expr 456	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂ_fld) ∧ 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
54	44, 46, 53	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
55	54	rexlimdva 3134	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → (∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
56		sstr2 3953	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
57	56	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
58	57	anim2d 612	. . . . . . . . . . . . . . 15 ⊢ ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
59	58	reximdv 3148	. . . . . . . . . . . . . 14 ⊢ ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → (∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
60	55, 59	syl9 77	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → (∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
61	60	impd 410	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 ∧ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
62	61	rexlimdva 3134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (∃𝑥 ∈ ℝ⁺ ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 ∧ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
63	39, 62	syl5 34	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((∃𝑥 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
64	63	expd 415	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (∃𝑥 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
65	38, 64	syl5 34	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝐶 ∈ 𝑢) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
66	65	expdimp 452	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂ_fld)) → (𝐶 ∈ 𝑢 → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
67	66	com23 86	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂ_fld)) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
68	67	ralrimdva 3133	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
69	36, 68	impbid 212	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
70	1	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → 𝐹:𝐴⟶ℂ)
71	70	ffund 6692	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → Fun 𝐹)
72		inss2 4201	. . . . . . . . . 10 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
73		difss 4099	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
74	70	fdmd 6698	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → dom 𝐹 = 𝐴)
75	73, 74	sseqtrrid 3990	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
76	72, 75	sstrid 3958	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
77		funimass4 6925	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
78	71, 76, 77	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
79	6	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (abs ∘ − ) ∈ (∞Met‘ℂ))
80		simplrr 777	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ⁺)
81	80	rpxrd 12996	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ^*)
82	3	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
83	73, 2	sstrid 3958	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
84	83	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
85	84	sselda 3946	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
86		elbl3 24280	. . . . . . . . . . . . 13 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℝ^*) ∧ (𝐵 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
87	79, 81, 82, 85, 86	syl22anc 838	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
88		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
89	88	cnmetdval 24658	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧 − 𝐵)))
90	85, 82, 89	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧 − 𝐵)))
91	90	breq1d 5117	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧(abs ∘ − )𝐵) < 𝑦 ↔ (abs‘(𝑧 − 𝐵)) < 𝑦))
92	87, 91	bitrd 279	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (abs‘(𝑧 − 𝐵)) < 𝑦))
93		simplrl 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ⁺)
94	93	rpxrd 12996	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ^*)
95		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐶 ∈ ℂ)
96		eldifi 4094	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ∈ 𝐴)
97		ffvelcdm 7053	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
98	70, 96, 97	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝐹‘𝑧) ∈ ℂ)
99		elbl3 24280	. . . . . . . . . . . . 13 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℝ^*) ∧ (𝐶 ∈ ℂ ∧ (𝐹‘𝑧) ∈ ℂ)) → ((𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹‘𝑧)(abs ∘ − )𝐶) < 𝑥))
100	79, 94, 95, 98, 99	syl22anc 838	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹‘𝑧)(abs ∘ − )𝐶) < 𝑥))
101	88	cnmetdval 24658	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑧) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹‘𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹‘𝑧) − 𝐶)))
102	98, 95, 101	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹‘𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹‘𝑧) − 𝐶)))
103	102	breq1d 5117	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (((𝐹‘𝑧)(abs ∘ − )𝐶) < 𝑥 ↔ (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))
104	100, 103	bitrd 279	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))
105	92, 104	imbi12d 344	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ ((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
106	105	ralbidva 3154	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (∀𝑧 ∈ (𝐴 ∖ {𝐵})(𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ ∀𝑧 ∈ (𝐴 ∖ {𝐵})((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
107		elin 3930	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})))
108	107	biancomi 462	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
109	108	imbi1i 349	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
110		impexp 450	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥))))
111	109, 110	bitr2i 276	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥))) ↔ (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
112	111	ralbii2 3071	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝐴 ∖ {𝐵})(𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ ∀𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
113		impexp 450	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝐵) → ((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧 ∈ 𝐴 → (𝑧 ≠ 𝐵 → ((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
114		eldifsn 4750	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝐵))
115	114	imbi1i 349	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝐵) → ((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
116		impexp 450	. . . . . . . . . . . 12 ⊢ (((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑧 ≠ 𝐵 → ((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
117	116	imbi2i 336	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 → ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧 ∈ 𝐴 → (𝑧 ≠ 𝐵 → ((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
118	113, 115, 117	3bitr4i 303	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧 ∈ 𝐴 → ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
119	118	ralbii2 3071	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝐴 ∖ {𝐵})((abs‘(𝑧 − 𝐵)) < 𝑦 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))
120	106, 112, 119	3bitr3g 313	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (∀𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
121	78, 120	bitrd 279	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
122	121	anassrs 467	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
123	122	rexbidva 3155	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → (∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
124	123	ralbidva 3154	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
125	69, 124	bitrd 279	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
126	125	pm5.32da 579	. 2 ⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
127	5, 126	bitrd 279	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 setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∖ cdif 3911 ∩ cin 3913 ⊆ wss 3914 {csn 4589 class class class wbr 5107 dom cdm 5638 “ cima 5641 ∘ ccom 5642 Fun wfun 6505 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝ^*cxr 11207 < clt 11208 − cmin 11405 ℝ⁺crp 12951 abscabs 15200 TopOpenctopn 17384 ∞Metcxmet 21249 ballcbl 21251 ℂ_fldccnfld 21264 lim_ℂ climc 25763

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fi 9362 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-fz 13469 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cnp 23115 df-xms 24208 df-ms 24209 df-limc 25767

This theorem is referenced by: dveflem 25883 dvferm1 25889 dvferm2 25891 lhop1 25919 ftc1lem6 25948 ulmdvlem3 26311 unblimceq0 36495 ftc1cnnc 37686 mullimc 45614 ellimcabssub0 45615 limcdm0 45616 mullimcf 45621 constlimc 45622 idlimc 45624 limcperiod 45626 limcrecl 45627 limcleqr 45642 neglimc 45645 addlimc 45646 0ellimcdiv 45647 limclner 45649 fperdvper 45917 ioodvbdlimc1lem2 45930 ioodvbdlimc2lem 45932

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