ellimc3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ellimc3		Structured version Visualization version GIF version

Theorem ellimc3 25838

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 2736	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
5	1, 2, 3, 4	ellimc2 25836	. 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 24718	. . . . . . . . 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 24359	. . . . . . . . 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 12917	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ^*)
12	11	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ^*)
13	4	cnfldtopn 24727	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blopn 24446	. . . . . . . . . 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 2825	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
17		sseq2 3960	. . . . . . . . . . . . 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 3160	. . . . . . . . . . 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 3572	. . . . . . . . 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 24439	. . . . . . . . . . 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 4194	. . . . . . . . . . . . 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 3940	. . . . . . . . . . . . 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 3151	. . . . . . . . . 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 3129	. . . . . . 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 3136	. . . . 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 24439	. . . . . . . . . 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 3100	. . . . . . . . . . 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 12952	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
43	13	blopn 24446	. . . . . . . . . . . . . . . . 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 24359	. . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
48		ineq1 4165	. . . . . . . . . . . . . . . . . . . . 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 3965	. . . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . . 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 3137	. . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . . . . 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 3151	. . . . . . . . . . . . . 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 3137	. . . . . . . . . . 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 3136	. . . . 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 6666	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → Fun 𝐹)
72		inss2 4190	. . . . . . . . . 10 ⊢ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
73		difss 4088	. . . . . . . . . . 11 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
74	70	fdmd 6672	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → dom 𝐹 = 𝐴)
75	73, 74	sseqtrrid 3977	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
76	72, 75	sstrid 3945	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
77		funimass4 6898	. . . . . . . . 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 12952	. . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
84	83	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
85	84	sselda 3933	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
86		elbl3 24338	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
89	88	cnmetdval 24716	. . . . . . . . . . . . . 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 5108	. . . . . . . . . . . 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 12952	. . . . . . . . . . . . 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 4083	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ∈ 𝐴)
97		ffvelcdm 7026	. . . . . . . . . . . . . 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 24338	. . . . . . . . . . . . 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 24716	. . . . . . . . . . . . . 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 5108	. . . . . . . . . . . 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 3157	. . . . . . . . 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 3917	. . . . . . . . . . . . 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 3078	. . . . . . . . 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 4742	. . . . . . . . . . . 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 3078	. . . . . . . . 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 3158	. . . . 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 3157	. . . 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 2932 ∀wral 3051 ∃wrex 3060 ∖ cdif 3898 ∩ cin 3900 ⊆ wss 3901 {csn 4580 class class class wbr 5098 dom cdm 5624 “ cima 5627 ∘ ccom 5628 Fun wfun 6486 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 ℝ^*cxr 11167 < clt 11168 − cmin 11366 ℝ⁺crp 12907 abscabs 15159 TopOpenctopn 17343 ∞Metcxmet 21296 ballcbl 21298 ℂ_fldccnfld 21311 lim_ℂ climc 25821

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8767 df-pm 8768 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fi 9316 df-sup 9347 df-inf 9348 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-fz 13426 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-rest 17344 df-topn 17345 df-topgen 17365 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cnp 23174 df-xms 24266 df-ms 24267 df-limc 25825

This theorem is referenced by: dveflem 25941 dvferm1 25947 dvferm2 25949 lhop1 25977 ftc1lem6 26006 ulmdvlem3 26369 unblimceq0 36709 ftc1cnnc 37895 mullimc 45883 ellimcabssub0 45884 limcdm0 45885 mullimcf 45890 constlimc 45891 idlimc 45893 limcperiod 45895 limcrecl 45896 limcleqr 45909 neglimc 45912 addlimc 45913 0ellimcdiv 45914 limclner 45916 fperdvper 46184 ioodvbdlimc1lem2 46197 ioodvbdlimc2lem 46199

W3C validator