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Theorem ellimc3 23449
 Description: Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.)
Hypotheses
Ref Expression
ellimc3.f (𝜑𝐹:𝐴⟶ℂ)
ellimc3.a (𝜑𝐴 ⊆ ℂ)
ellimc3.b (𝜑𝐵 ∈ ℂ)
Assertion
Ref Expression
ellimc3 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 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.
StepHypRef Expression
1 ellimc3.f . . 3 (𝜑𝐹:𝐴⟶ℂ)
2 ellimc3.a . . 3 (𝜑𝐴 ⊆ ℂ)
3 ellimc3.b . . 3 (𝜑𝐵 ∈ ℂ)
4 eqid 2610 . . 3 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
51, 2, 3, 4ellimc2 23447 . 2 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
6 cnxmet 22386 . . . . . . . . . 10 (abs ∘ − ) ∈ (∞Met‘ℂ)
76a1i 11 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (abs ∘ − ) ∈ (∞Met‘ℂ))
8 simplr 788 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ ℂ)
9 simpr 476 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
10 blcntr 22028 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
117, 8, 9, 10syl3anc 1318 . . . . . . . 8 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
12 rpxr 11716 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
1312adantl 481 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ*)
144cnfldtopn 22395 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
1514blopn 22115 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ*) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂfld))
167, 8, 13, 15syl3anc 1318 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂfld))
17 eleq2 2677 . . . . . . . . . . 11 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶𝑢𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
18 sseq2 3590 . . . . . . . . . . . . 13 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
1918anbi2d 736 . . . . . . . . . . . 12 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
2019rexbidv 3034 . . . . . . . . . . 11 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
2117, 20imbi12d 333 . . . . . . . . . 10 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) ↔ (𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))))
2221rspcv 3278 . . . . . . . . 9 ((𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂfld) → (∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → (𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))))
2316, 22syl 17 . . . . . . . 8 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → (𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))))
2411, 23mpid 43 . . . . . . 7 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
2514mopni2 22108 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ∃𝑦 ∈ ℝ+ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
266, 25mp3an1 1403 . . . . . . . . . 10 ((𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ∃𝑦 ∈ ℝ+ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
27 ssrin 3800 . . . . . . . . . . . . 13 ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
28 imass2 5420 . . . . . . . . . . . . 13 (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
29 sstr2 3575 . . . . . . . . . . . . 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)))
3027, 28, 293syl 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)))
3130com12 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)))
3231reximdv 2999 . . . . . . . . . 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)))
3326, 32syl5com 31 . . . . . . . . 9 ((𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
3433impr 647 . . . . . . . 8 ((𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))) → ∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))
3534rexlimiva 3010 . . . . . . 7 (∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))
3624, 35syl6 34 . . . . . 6 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
3736ralrimdva 2952 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
3814mopni2 22108 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → ∃𝑥 ∈ ℝ+ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
396, 38mp3an1 1403 . . . . . . . . 9 ((𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → ∃𝑥 ∈ ℝ+ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
40 r19.29r 3055 . . . . . . . . . . 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)))
416a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (abs ∘ − ) ∈ (∞Met‘ℂ))
423ad3antrrr 762 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ ℂ)
43 simpr 476 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
4443rpxrd 11749 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ*)
4514blopn 22115 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ*) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld))
4641, 42, 44, 45syl3anc 1318 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld))
47 blcntr 22028 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
4841, 42, 43, 47syl3anc 1318 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
49 eleq2 2677 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵𝑣𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
50 ineq1 3769 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
5150imaeq2d 5385 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
5251sseq1d 3595 . . . . . . . . . . . . . . . . . . 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)))
5349, 52anbi12d 743 . . . . . . . . . . . . . . . . . 18 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ (𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
5453rspcev 3282 . . . . . . . . . . . . . . . . 17 (((𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
5554expr 641 . . . . . . . . . . . . . . . 16 (((𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld) ∧ 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
5646, 48, 55syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
5756rexlimdva 3013 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
58 sstr2 3575 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
5958com12 32 . . . . . . . . . . . . . . . 16 ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
6059anim2d 587 . . . . . . . . . . . . . . 15 ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
6160reximdv 2999 . . . . . . . . . . . . . 14 ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → (∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
6257, 61syl9 75 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → (∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
6362impd 446 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 ∧ ∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
6463rexlimdva 3013 . . . . . . . . . . 11 ((𝜑𝐶 ∈ ℂ) → (∃𝑥 ∈ ℝ+ ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 ∧ ∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
6540, 64syl5 33 . . . . . . . . . 10 ((𝜑𝐶 ∈ 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‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
6665expd 451 . . . . . . . . 9 ((𝜑𝐶 ∈ 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‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
6739, 66syl5 33 . . . . . . . 8 ((𝜑𝐶 ∈ ℂ) → ((𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
6867expdimp 452 . . . . . . 7 (((𝜑𝐶 ∈ ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
6968com23 84 . . . . . 6 (((𝜑𝐶 ∈ ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
7069ralrimdva 2952 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
7137, 70impbid 201 . . . 4 ((𝜑𝐶 ∈ ℂ) → (∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
721ad2antrr 758 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → 𝐹:𝐴⟶ℂ)
73 ffun 5961 . . . . . . . . . 10 (𝐹:𝐴⟶ℂ → Fun 𝐹)
7472, 73syl 17 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → Fun 𝐹)
75 inss2 3796 . . . . . . . . . 10 ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
76 difss 3699 . . . . . . . . . . 11 (𝐴 ∖ {𝐵}) ⊆ 𝐴
77 fdm 5964 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
7872, 77syl 17 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → dom 𝐹 = 𝐴)
7976, 78syl5sseqr 3617 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
8075, 79syl5ss 3579 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
81 funimass4 6157 . . . . . . . . 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}))(𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
8274, 80, 81syl2anc 691 . . . . . . . 8 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
836a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (abs ∘ − ) ∈ (∞Met‘ℂ))
84 simplrr 797 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ+)
8584rpxrd 11749 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ*)
863ad3antrrr 762 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
8776, 2syl5ss 3579 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8887ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8988sselda 3568 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
90 elbl3 22007 . . . . . . . . . . . . 13 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℝ*) ∧ (𝐵 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
9183, 85, 86, 89, 90syl22anc 1319 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
92 eqid 2610 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
9392cnmetdval 22384 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9489, 86, 93syl2anc 691 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9594breq1d 4593 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧(abs ∘ − )𝐵) < 𝑦 ↔ (abs‘(𝑧𝐵)) < 𝑦))
9691, 95bitrd 267 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (abs‘(𝑧𝐵)) < 𝑦))
97 simplrl 796 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ+)
9897rpxrd 11749 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ*)
99 simpllr 795 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐶 ∈ ℂ)
100 eldifi 3694 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐴)
101 ffvelrn 6265 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
10272, 100, 101syl2an 493 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝐹𝑧) ∈ ℂ)
103 elbl3 22007 . . . . . . . . . . . . 13 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℝ*) ∧ (𝐶 ∈ ℂ ∧ (𝐹𝑧) ∈ ℂ)) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥))
10483, 98, 99, 102, 103syl22anc 1319 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥))
10592cnmetdval 22384 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
106102, 99, 105syl2anc 691 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
107106breq1d 4593 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥 ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
108104, 107bitrd 267 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
10996, 108imbi12d 333 . . . . . . . . . 10 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
110109ralbidva 2968 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (∀𝑧 ∈ (𝐴 ∖ {𝐵})(𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ ∀𝑧 ∈ (𝐴 ∖ {𝐵})((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
111 elin 3758 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})))
112 ancom 465 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
113111, 112bitri 263 . . . . . . . . . . . 12 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
114113imbi1i 338 . . . . . . . . . . 11 ((𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
115 impexp 461 . . . . . . . . . . 11 (((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥))))
116114, 115bitr2i 264 . . . . . . . . . 10 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥))) ↔ (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
117116ralbii2 2961 . . . . . . . . 9 (∀𝑧 ∈ (𝐴 ∖ {𝐵})(𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥)) ↔ ∀𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
118 impexp 461 . . . . . . . . . . 11 (((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
119 eldifsn 4260 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧𝐴𝑧𝐵))
120119imbi1i 338 . . . . . . . . . . 11 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ ((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
121 impexp 461 . . . . . . . . . . . 12 (((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
122121imbi2i 325 . . . . . . . . . . 11 ((𝑧𝐴 → ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
123118, 120, 1223bitr4i 291 . . . . . . . . . 10 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
124123ralbii2 2961 . . . . . . . . 9 (∀𝑧 ∈ (𝐴 ∖ {𝐵})((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
125110, 117, 1243bitr3g 301 . . . . . . . 8 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (∀𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
12682, 125bitrd 267 . . . . . . 7 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
127126anassrs 678 . . . . . 6 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ((𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
128127rexbidva 3031 . . . . 5 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∃𝑦 ∈ ℝ+𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
129128ralbidva 2968 . . . 4 ((𝜑𝐶 ∈ ℂ) → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+ (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
13071, 129bitrd 267 . . 3 ((𝜑𝐶 ∈ ℂ) → (∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
131130pm5.32da 671 . 2 (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
1325, 131bitrd 267 1 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125   class class class wbr 4583  dom cdm 5038   “ cima 5041   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝ*cxr 9952   < clt 9953   − cmin 10145  ℝ+crp 11708  abscabs 13822  TopOpenctopn 15905  ∞Metcxmt 19552  ballcbl 19554  ℂfldccnfld 19567   limℂ climc 23432 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-topn 15907  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cnp 20842  df-xms 21935  df-ms 21936  df-limc 23436 This theorem is referenced by:  dveflem  23546  dvferm1  23552  dvferm2  23554  lhop1  23581  ftc1lem6  23608  ulmdvlem3  23960  unblimceq0  31668  ftc1cnnc  32654  mullimc  38683  ellimcabssub0  38684  limcdm0  38685  mullimcf  38690  constlimc  38691  idlimc  38693  limcperiod  38695  limcrecl  38696  limcleqr  38711  neglimc  38714  addlimc  38715  0ellimcdiv  38716  limclner  38718  fperdvper  38808  ioodvbdlimc1lem2  38822  ioodvbdlimc2lem  38824
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