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Theorem ellimc3 25834

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 2734	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	1, 2, 3, 4	ellimc2 25832	. 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 24714	. . . . . . . . 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 24355	. . . . . . . . 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 12913	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ^*)
12	11	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ^*)
13	4	cnfldtopn 24723	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blopn 24442	. . . . . . . . . 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 2823	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
17		sseq2 3958	. . . . . . . . . . . . 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 3158	. . . . . . . . . . 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 3570	. . . . . . . . 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 24435	. . . . . . . . . . 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 4192	. . . . . . . . . . . . 13 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
27		imass2 6059	. . . . . . . . . . . . 13 ⊢ (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
28		sstr2 3938	. . . . . . . . . . . . 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 3149	. . . . . . . . . 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 3127	. . . . . . 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 3134	. . . . 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 24435	. . . . . . . . . 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 3098	. . . . . . . . . . 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 12948	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
43	13	blopn 24442	. . . . . . . . . . . . . . . . 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 24355	. . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
48		ineq1 4163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
49	48	imaeq2d 6017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
50	49	sseq1d 3963	. . . . . . . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . . . . . 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 3135	. . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . 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 3135	. . . . . . . . . . 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 3134	. . . . 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 6664	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → Fun 𝐹)
72		inss2 4188	. . . . . . . . . 10 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
73		difss 4086	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
74	70	fdmd 6670	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → dom 𝐹 = 𝐴)
75	73, 74	sseqtrrid 3975	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
76	72, 75	sstrid 3943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
77		funimass4 6896	. . . . . . . . 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 12948	. . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
84	83	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
85	84	sselda 3931	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
86		elbl3 24334	. . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
89	88	cnmetdval 24712	. . . . . . . . . . . . . 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 5106	. . . . . . . . . . . 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 12948	. . . . . . . . . . . . 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 4081	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ∈ 𝐴)
97		ffvelcdm 7024	. . . . . . . . . . . . . 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 24334	. . . . . . . . . . . . 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 24712	. . . . . . . . . . . . . 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 5106	. . . . . . . . . . . 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 3155	. . . . . . . . 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 3915	. . . . . . . . . . . . 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 3076	. . . . . . . . 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 4740	. . . . . . . . . . . 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 3076	. . . . . . . . 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 3156	. . . . 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 3155	. . . 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 1541 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 ∃wrex 3058 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 {csn 4578 class class class wbr 5096 dom cdm 5622 “ cima 5625 ∘ ccom 5626 Fun wfun 6484 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝ^*cxr 11163 < clt 11164 − cmin 11362 ℝ⁺crp 12903 abscabs 15155 TopOpenctopn 17339 ∞Metcxmet 21292 ballcbl 21294 ℂ_fldccnfld 21307 lim_ℂ climc 25817

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fi 9312 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-fz 13422 df-seq 13923 df-exp 13983 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-struct 17072 df-slot 17107 df-ndx 17119 df-base 17135 df-plusg 17188 df-mulr 17189 df-starv 17190 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-rest 17340 df-topn 17341 df-topgen 17361 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cnp 23170 df-xms 24262 df-ms 24263 df-limc 25821

This theorem is referenced by: dveflem 25937 dvferm1 25943 dvferm2 25945 lhop1 25973 ftc1lem6 26002 ulmdvlem3 26365 unblimceq0 36650 ftc1cnnc 37832 mullimc 45804 ellimcabssub0 45805 limcdm0 45806 mullimcf 45811 constlimc 45812 idlimc 45814 limcperiod 45816 limcrecl 45817 limcleqr 45830 neglimc 45833 addlimc 45834 0ellimcdiv 45835 limclner 45837 fperdvper 46105 ioodvbdlimc1lem2 46118 ioodvbdlimc2lem 46120

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