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Theorem ellimc3 24631
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 2738 . . 3 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
51, 2, 3, 4ellimc2 24629 . 2 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
6 cnxmet 23526 . . . . . . . . 9 (abs ∘ − ) ∈ (∞Met‘ℂ)
7 simplr 769 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ ℂ)
8 simpr 488 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
9 blcntr 23167 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
106, 7, 8, 9mp3an2i 1467 . . . . . . . 8 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
11 rpxr 12482 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
1211adantl 485 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ*)
134cnfldtopn 23535 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
1413blopn 23254 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ*) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂfld))
156, 7, 12, 14mp3an2i 1467 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂfld))
16 eleq2 2821 . . . . . . . . . . 11 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶𝑢𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
17 sseq2 3904 . . . . . . . . . . . . 13 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
1817anbi2d 632 . . . . . . . . . . . 12 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
1918rexbidv 3207 . . . . . . . . . . 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))))
2016, 19imbi12d 348 . . . . . . . . . 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)))))
2120rspcv 3522 . . . . . . . . 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)))))
2215, 21syl 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)))))
2310, 22mpid 44 . . . . . . 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))))
2413mopni2 23247 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ∃𝑦 ∈ ℝ+ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
256, 24mp3an1 1449 . . . . . . . . . 10 ((𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ∃𝑦 ∈ ℝ+ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
26 ssrin 4125 . . . . . . . . . . . . 13 ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
27 imass2 5940 . . . . . . . . . . . . 13 (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
28 sstr2 3885 . . . . . . . . . . . . 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)))
2926, 27, 283syl 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)))
3029com12 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)))
3130reximdv 3183 . . . . . . . . . 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)))
3225, 31syl5com 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)))
3332impr 458 . . . . . . . 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))
3433rexlimiva 3191 . . . . . . 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))
3523, 34syl6 35 . . . . . 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)))
3635ralrimdva 3101 . . . . 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)))
3713mopni2 23247 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → ∃𝑥 ∈ ℝ+ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
386, 37mp3an1 1449 . . . . . . . . 9 ((𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → ∃𝑥 ∈ ℝ+ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
39 r19.29r 3168 . . . . . . . . . . 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)))
403ad3antrrr 730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ ℂ)
41 simpr 488 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
4241rpxrd 12516 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ*)
4313blopn 23254 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ*) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld))
446, 40, 42, 43mp3an2i 1467 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld))
45 blcntr 23167 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
466, 40, 41, 45mp3an2i 1467 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
47 eleq2 2821 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵𝑣𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
48 ineq1 4097 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
4948imaeq2d 5904 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
5049sseq1d 3909 . . . . . . . . . . . . . . . . . . 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)))
5147, 50anbi12d 634 . . . . . . . . . . . . . . . . . 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))))
5251rspcev 3527 . . . . . . . . . . . . . . . . 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)))
5352expr 460 . . . . . . . . . . . . . . . 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))))
5444, 46, 53syl2anc 587 . . . . . . . . . . . . . . 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))))
5554rexlimdva 3194 . . . . . . . . . . . . . 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))))
56 sstr2 3885 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
5756com12 32 . . . . . . . . . . . . . . . 16 ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
5857anim2d 615 . . . . . . . . . . . . . . 15 ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
5958reximdv 3183 . . . . . . . . . . . . . 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)))
6055, 59syl9 77 . . . . . . . . . . . . 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))))
6160impd 414 . . . . . . . . . . . 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)))
6261rexlimdva 3194 . . . . . . . . . . 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)))
6339, 62syl5 34 . . . . . . . . . 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)))
6463expd 419 . . . . . . . . 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))))
6538, 64syl5 34 . . . . . . . 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))))
6665expdimp 456 . . . . . . 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))))
6766com23 86 . . . . . 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))))
6867ralrimdva 3101 . . . . 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))))
6936, 68impbid 215 . . . 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)))
701ad2antrr 726 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → 𝐹:𝐴⟶ℂ)
7170ffund 6509 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → Fun 𝐹)
72 inss2 4121 . . . . . . . . . 10 ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
73 difss 4023 . . . . . . . . . . 11 (𝐴 ∖ {𝐵}) ⊆ 𝐴
7470fdmd 6516 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → dom 𝐹 = 𝐴)
7573, 74sseqtrrid 3931 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
7672, 75sstrid 3889 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
77 funimass4 6735 . . . . . . . . 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)))
7871, 76, 77syl2anc 587 . . . . . . . 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)))
796a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (abs ∘ − ) ∈ (∞Met‘ℂ))
80 simplrr 778 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ+)
8180rpxrd 12516 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ*)
823ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
8373, 2sstrid 3889 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8483ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8584sselda 3878 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
86 elbl3 23146 . . . . . . . . . . . . 13 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℝ*) ∧ (𝐵 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
8779, 81, 82, 85, 86syl22anc 838 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
88 eqid 2738 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
8988cnmetdval 23524 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9085, 82, 89syl2anc 587 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9190breq1d 5041 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧(abs ∘ − )𝐵) < 𝑦 ↔ (abs‘(𝑧𝐵)) < 𝑦))
9287, 91bitrd 282 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (abs‘(𝑧𝐵)) < 𝑦))
93 simplrl 777 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ+)
9493rpxrd 12516 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ*)
95 simpllr 776 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐶 ∈ ℂ)
96 eldifi 4018 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐴)
97 ffvelrn 6860 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
9870, 96, 97syl2an 599 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝐹𝑧) ∈ ℂ)
99 elbl3 23146 . . . . . . . . . . . . 13 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℝ*) ∧ (𝐶 ∈ ℂ ∧ (𝐹𝑧) ∈ ℂ)) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥))
10079, 94, 95, 98, 99syl22anc 838 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥))
10188cnmetdval 23524 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
10298, 95, 101syl2anc 587 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
103102breq1d 5041 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥 ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
104100, 103bitrd 282 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
10592, 104imbi12d 348 . . . . . . . . . 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)))
106105ralbidva 3108 . . . . . . . . 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)))
107 elin 3860 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})))
108107biancomi 466 . . . . . . . . . . . 12 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
109108imbi1i 353 . . . . . . . . . . 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)))
110 impexp 454 . . . . . . . . . . 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))))
111109, 110bitr2i 279 . . . . . . . . . 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)))
112111ralbii2 3078 . . . . . . . . 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))
113 impexp 454 . . . . . . . . . . 11 (((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
114 eldifsn 4676 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧𝐴𝑧𝐵))
115114imbi1i 353 . . . . . . . . . . 11 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ ((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
116 impexp 454 . . . . . . . . . . . 12 (((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
117116imbi2i 339 . . . . . . . . . . 11 ((𝑧𝐴 → ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
118113, 115, 1173bitr4i 306 . . . . . . . . . 10 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
119118ralbii2 3078 . . . . . . . . 9 (∀𝑧 ∈ (𝐴 ∖ {𝐵})((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
120106, 112, 1193bitr3g 316 . . . . . . . 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)))
12178, 120bitrd 282 . . . . . . 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)))
122121anassrs 471 . . . . . 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)))
123122rexbidva 3206 . . . . 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)))
124123ralbidva 3108 . . . 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)))
12569, 124bitrd 282 . . 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)))
126125pm5.32da 582 . 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))))
1275, 126bitrd 282 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 209  wa 399   = wceq 1542  wcel 2113  wne 2934  wral 3053  wrex 3054  cdif 3841  cin 3843  wss 3844  {csn 4517   class class class wbr 5031  dom cdm 5526  cima 5529  ccom 5530  Fun wfun 6334  wf 6336  cfv 6340  (class class class)co 7171  cc 10614  *cxr 10753   < clt 10754  cmin 10949  +crp 12473  abscabs 14684  TopOpenctopn 16799  ∞Metcxmet 20203  ballcbl 20205  fldccnfld 20218   lim climc 24614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-cnex 10672  ax-resscn 10673  ax-1cn 10674  ax-icn 10675  ax-addcl 10676  ax-addrcl 10677  ax-mulcl 10678  ax-mulrcl 10679  ax-mulcom 10680  ax-addass 10681  ax-mulass 10682  ax-distr 10683  ax-i2m1 10684  ax-1ne0 10685  ax-1rid 10686  ax-rnegex 10687  ax-rrecex 10688  ax-cnre 10689  ax-pre-lttri 10690  ax-pre-lttrn 10691  ax-pre-ltadd 10692  ax-pre-mulgt0 10693  ax-pre-sup 10694
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7128  df-ov 7174  df-oprab 7175  df-mpo 7176  df-om 7601  df-1st 7715  df-2nd 7716  df-wrecs 7977  df-recs 8038  df-rdg 8076  df-1o 8132  df-er 8321  df-map 8440  df-pm 8441  df-en 8557  df-dom 8558  df-sdom 8559  df-fin 8560  df-fi 8949  df-sup 8980  df-inf 8981  df-pnf 10756  df-mnf 10757  df-xr 10758  df-ltxr 10759  df-le 10760  df-sub 10951  df-neg 10952  df-div 11377  df-nn 11718  df-2 11780  df-3 11781  df-4 11782  df-5 11783  df-6 11784  df-7 11785  df-8 11786  df-9 11787  df-n0 11978  df-z 12064  df-dec 12181  df-uz 12326  df-q 12432  df-rp 12474  df-xneg 12591  df-xadd 12592  df-xmul 12593  df-fz 12983  df-seq 13462  df-exp 13523  df-cj 14549  df-re 14550  df-im 14551  df-sqrt 14685  df-abs 14686  df-struct 16589  df-ndx 16590  df-slot 16591  df-base 16593  df-plusg 16682  df-mulr 16683  df-starv 16684  df-tset 16688  df-ple 16689  df-ds 16691  df-unif 16692  df-rest 16800  df-topn 16801  df-topgen 16821  df-psmet 20210  df-xmet 20211  df-met 20212  df-bl 20213  df-mopn 20214  df-cnfld 20219  df-top 21646  df-topon 21663  df-topsp 21685  df-bases 21698  df-cnp 21980  df-xms 23074  df-ms 23075  df-limc 24618
This theorem is referenced by:  dveflem  24731  dvferm1  24737  dvferm2  24739  lhop1  24766  ftc1lem6  24793  ulmdvlem3  25149  unblimceq0  34325  ftc1cnnc  35469  mullimc  42691  ellimcabssub0  42692  limcdm0  42693  mullimcf  42698  constlimc  42699  idlimc  42701  limcperiod  42703  limcrecl  42704  limcleqr  42719  neglimc  42722  addlimc  42723  0ellimcdiv  42724  limclner  42726  fperdvper  42994  ioodvbdlimc1lem2  43007  ioodvbdlimc2lem  43009
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