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Theorem ellimc3 25807

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 2731	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	1, 2, 3, 4	ellimc2 25805	. 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 24687	. . . . . . . . 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 24328	. . . . . . . . 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 12900	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ^*)
12	11	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ^*)
13	4	cnfldtopn 24696	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blopn 24415	. . . . . . . . . 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 2820	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
17		sseq2 3956	. . . . . . . . . . . . 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 3156	. . . . . . . . . . 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 3568	. . . . . . . . 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 24408	. . . . . . . . . . 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 4189	. . . . . . . . . . . . 13 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
27		imass2 6050	. . . . . . . . . . . . 13 ⊢ (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
28		sstr2 3936	. . . . . . . . . . . . 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 3147	. . . . . . . . . 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 3125	. . . . . . 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 3132	. . . . 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 24408	. . . . . . . . . 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 12935	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
43	13	blopn 24415	. . . . . . . . . . . . . . . . 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 24328	. . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
48		ineq1 4160	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
49	48	imaeq2d 6008	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
50	49	sseq1d 3961	. . . . . . . . . . . . . . . . . . 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 3572	. . . . . . . . . . . . . . . . 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 3133	. . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . 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 3133	. . . . . . . . . . 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 3132	. . . . 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 6655	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → Fun 𝐹)
72		inss2 4185	. . . . . . . . . 10 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
73		difss 4083	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
74	70	fdmd 6661	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → dom 𝐹 = 𝐴)
75	73, 74	sseqtrrid 3973	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
76	72, 75	sstrid 3941	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
77		funimass4 6886	. . . . . . . . 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 12935	. . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
84	83	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
85	84	sselda 3929	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
86		elbl3 24307	. . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
89	88	cnmetdval 24685	. . . . . . . . . . . . . 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 5099	. . . . . . . . . . . 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 12935	. . . . . . . . . . . . 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 4078	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ∈ 𝐴)
97		ffvelcdm 7014	. . . . . . . . . . . . . 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 24307	. . . . . . . . . . . . 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 24685	. . . . . . . . . . . . . 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 5099	. . . . . . . . . . . 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 3153	. . . . . . . . 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 3913	. . . . . . . . . . . . 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 3074	. . . . . . . . 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 4735	. . . . . . . . . . . 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 3074	. . . . . . . . 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 3154	. . . . 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 3153	. . . 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 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 ∖ cdif 3894 ∩ cin 3896 ⊆ wss 3897 {csn 4573 class class class wbr 5089 dom cdm 5614 “ cima 5617 ∘ ccom 5618 Fun wfun 6475 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝ^*cxr 11145 < clt 11146 − cmin 11344 ℝ⁺crp 12890 abscabs 15141 TopOpenctopn 17325 ∞Metcxmet 21276 ballcbl 21278 ℂ_fldccnfld 21291 lim_ℂ climc 25790

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-fz 13408 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-rest 17326 df-topn 17327 df-topgen 17347 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cnp 23143 df-xms 24235 df-ms 24236 df-limc 25794

This theorem is referenced by: dveflem 25910 dvferm1 25916 dvferm2 25918 lhop1 25946 ftc1lem6 25975 ulmdvlem3 26338 unblimceq0 36551 ftc1cnnc 37742 mullimc 45726 ellimcabssub0 45727 limcdm0 45728 mullimcf 45733 constlimc 45734 idlimc 45736 limcperiod 45738 limcrecl 45739 limcleqr 45752 neglimc 45755 addlimc 45756 0ellimcdiv 45757 limclner 45759 fperdvper 46027 ioodvbdlimc1lem2 46040 ioodvbdlimc2lem 46042

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