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Theorem ellimc3 25868

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 2741	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	1, 2, 3, 4	ellimc2 25866	. 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 24759	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
7		simplr 775	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
8		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
9		blcntr 24400	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
10	6, 7, 8, 9	mp3an2i 1475	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
11		rpxr 12947	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ^*)
12	11	adantl 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ^*)
13	4	cnfldtopn 24768	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blopn 24487	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ^*) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂ_fld))
15	6, 7, 12, 14	mp3an2i 1475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂ_fld))
16		eleq2 2830	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
17		sseq2 3943	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
18	17	anbi2d 637	. . . . . . . . . . . 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 3165	. . . . . . . . . . 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 346	. . . . . . . . . 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 3558	. . . . . . . . 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 24480	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑣 ∈ (TopOpen‘ℂ_fld) ∧ 𝐵 ∈ 𝑣) → ∃𝑦 ∈ ℝ⁺ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
25	6, 24	mp3an1 1457	. . . . . . . . . 10 ⊢ ((𝑣 ∈ (TopOpen‘ℂ_fld) ∧ 𝐵 ∈ 𝑣) → ∃𝑦 ∈ ℝ⁺ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
26		ssrin 4173	. . . . . . . . . . . . 13 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
27		imass2 6061	. . . . . . . . . . . . 13 ⊢ (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
28		sstr2 3924	. . . . . . . . . . . . 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 3156	. . . . . . . . . 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 456	. . . . . . . 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 3134	. . . . . . 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 3141	. . . . 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 24480	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝐶 ∈ 𝑢) → ∃𝑥 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
38	6, 37	mp3an1 1457	. . . . . . . . 9 ⊢ ((𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝐶 ∈ 𝑢) → ∃𝑥 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
39		r19.29r 3105	. . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
41		simpr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
42	41	rpxrd 12982	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
43	13	blopn 24487	. . . . . . . . . . . . . . . . 17 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ^*) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂ_fld))
44	6, 40, 42, 43	mp3an2i 1475	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂ_fld))
45		blcntr 24400	. . . . . . . . . . . . . . . . 17 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ⁺) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
46	6, 40, 41, 45	mp3an2i 1475	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
47		eleq2 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
48		ineq1 4145	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
49	48	imaeq2d 6019	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
50	49	sseq1d 3948	. . . . . . . . . . . . . . . . . . 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 639	. . . . . . . . . . . . . . . . . 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 3562	. . . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . 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 3142	. . . . . . . . . . . . . 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 3924	. . . . . . . . . . . . . . . . 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 619	. . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 3142	. . . . . . . . . . 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 417	. . . . . . . . 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 454	. . . . . . 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 3141	. . . . 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 214	. . . 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 733	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → 𝐹:𝐴⟶ℂ)
71	70	ffund 6663	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → Fun 𝐹)
72		inss2 4169	. . . . . . . . . 10 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
73		difss 4069	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
74	70	fdmd 6669	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → dom 𝐹 = 𝐴)
75	73, 74	sseqtrrid 3960	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
76	72, 75	sstrid 3928	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
77		funimass4 6895	. . . . . . . . 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 591	. . . . . . . 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 784	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ⁺)
81	80	rpxrd 12982	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ^*)
82	3	ad3antrrr 737	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
83	73, 2	sstrid 3928	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
84	83	ad2antrr 733	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
85	84	sselda 3917	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
86		elbl3 24379	. . . . . . . . . . . . 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 845	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
88		eqid 2741	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
89	88	cnmetdval 24757	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧 − 𝐵)))
90	85, 82, 89	syl2anc 591	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧 − 𝐵)))
91	90	breq1d 5085	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧(abs ∘ − )𝐵) < 𝑦 ↔ (abs‘(𝑧 − 𝐵)) < 𝑦))
92	87, 91	bitrd 281	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (abs‘(𝑧 − 𝐵)) < 𝑦))
93		simplrl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ⁺)
94	93	rpxrd 12982	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ^*)
95		simpllr 782	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐶 ∈ ℂ)
96		eldifi 4064	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ∈ 𝐴)
97		ffvelcdm 7026	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
98	70, 96, 97	syl2an 603	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝐹‘𝑧) ∈ ℂ)
99		elbl3 24379	. . . . . . . . . . . . 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 845	. . . . . . . . . . . 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 24757	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑧) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹‘𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹‘𝑧) − 𝐶)))
102	98, 95, 101	syl2anc 591	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹‘𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹‘𝑧) − 𝐶)))
103	102	breq1d 5085	. . . . . . . . . . . 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 281	. . . . . . . . . . 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 346	. . . . . . . . . 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 3162	. . . . . . . . 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 3901	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})))
108	107	biancomi 464	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
109	108	imbi1i 351	. . . . . . . . . . 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 452	. . . . . . . . . . 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 278	. . . . . . . . . 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 3083	. . . . . . . . 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 452	. . . . . . . . . . 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 4722	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝐵))
115	114	imbi1i 351	. . . . . . . . . . 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 452	. . . . . . . . . . . 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 338	. . . . . . . . . . 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 305	. . . . . . . . . 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 3083	. . . . . . . . 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 315	. . . . . . . 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 281	. . . . . . 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 469	. . . . . 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 3163	. . . . 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 3162	. . . 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 281	. . 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 585	. 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 281	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 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 ∃wrex 3065 ∖ cdif 3882 ∩ cin 3884 ⊆ wss 3885 {csn 4558 class class class wbr 5075 dom cdm 5621 “ cima 5624 ∘ ccom 5625 Fun wfun 6483 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 ℂcc 11031 ℝ^*cxr 11173 < clt 11174 − cmin 11372 ℝ⁺crp 12937 abscabs 15191 TopOpenctopn 17379 ∞Metcxmet 21336 ballcbl 21338 ℂ_fldccnfld 21351 lim_ℂ climc 25851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fi 9318 df-sup 9349 df-inf 9350 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-fz 13457 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-plusg 17228 df-mulr 17229 df-starv 17230 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-rest 17380 df-topn 17381 df-topgen 17401 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cnp 23215 df-xms 24307 df-ms 24308 df-limc 25855

This theorem is referenced by: dveflem 25968 dvferm1 25974 dvferm2 25976 lhop1 26003 ftc1lem6 26030 ulmdvlem3 26389 unblimceq0 36828 ftc1cnnc 38074 mullimc 46075 ellimcabssub0 46076 limcdm0 46077 mullimcf 46082 constlimc 46083 idlimc 46085 limcperiod 46087 limcrecl 46088 limcleqr 46101 neglimc 46104 addlimc 46105 0ellimcdiv 46106 limclner 46108 fperdvper 46376 ioodvbdlimc1lem2 46389 ioodvbdlimc2lem 46391

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