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Theorem ellimc3 25727
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 2731 . . 3 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
51, 2, 3, 4ellimc2 25725 . 2 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
6 cnxmet 24608 . . . . . . . . 9 (abs ∘ − ) ∈ (∞Met‘ℂ)
7 simplr 766 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ ℂ)
8 simpr 484 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
9 blcntr 24238 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
106, 7, 8, 9mp3an2i 1465 . . . . . . . 8 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
11 rpxr 12990 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
1211adantl 481 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ*)
134cnfldtopn 24617 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
1413blopn 24328 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ*) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂfld))
156, 7, 12, 14mp3an2i 1465 . . . . . . . . 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 4008 . . . . . . . . . . . . 13 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
1817anbi2d 628 . . . . . . . . . . . 12 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
1918rexbidv 3177 . . . . . . . . . . 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 344 . . . . . . . . . 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 3608 . . . . . . . . 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 24321 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ∃𝑦 ∈ ℝ+ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
256, 24mp3an1 1447 . . . . . . . . . 10 ((𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ∃𝑦 ∈ ℝ+ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
26 ssrin 4233 . . . . . . . . . . . . 13 ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
27 imass2 6101 . . . . . . . . . . . . 13 (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
28 sstr2 3989 . . . . . . . . . . . . 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 3169 . . . . . . . . . 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 454 . . . . . . . 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 3146 . . . . . . 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 3153 . . . . 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 24321 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → ∃𝑥 ∈ ℝ+ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
386, 37mp3an1 1447 . . . . . . . . 9 ((𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → ∃𝑥 ∈ ℝ+ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
39 r19.29r 3115 . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ ℂ)
41 simpr 484 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
4241rpxrd 13024 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ*)
4313blopn 24328 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ*) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld))
446, 40, 42, 43mp3an2i 1465 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld))
45 blcntr 24238 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
466, 40, 41, 45mp3an2i 1465 . . . . . . . . . . . . . . . 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 4205 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
4948imaeq2d 6059 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
5049sseq1d 4013 . . . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . . . 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 3612 . . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . 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 3154 . . . . . . . . . . . . . 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 3989 . . . . . . . . . . . . . . . . 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 611 . . . . . . . . . . . . . . 15 ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
5958reximdv 3169 . . . . . . . . . . . . . 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 410 . . . . . . . . . . . 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 3154 . . . . . . . . . . 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 415 . . . . . . . . 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 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))))
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 3153 . . . . 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 211 . . . 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 723 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → 𝐹:𝐴⟶ℂ)
7170ffund 6721 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → Fun 𝐹)
72 inss2 4229 . . . . . . . . . 10 ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
73 difss 4131 . . . . . . . . . . 11 (𝐴 ∖ {𝐵}) ⊆ 𝐴
7470fdmd 6728 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → dom 𝐹 = 𝐴)
7573, 74sseqtrrid 4035 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
7672, 75sstrid 3993 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
77 funimass4 6956 . . . . . . . . 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 583 . . . . . . . 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 775 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ+)
8180rpxrd 13024 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ*)
823ad3antrrr 727 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
8373, 2sstrid 3993 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8483ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8584sselda 3982 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
86 elbl3 24217 . . . . . . . . . . . . 13 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℝ*) ∧ (𝐵 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
8779, 81, 82, 85, 86syl22anc 836 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
88 eqid 2731 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
8988cnmetdval 24606 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9085, 82, 89syl2anc 583 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9190breq1d 5158 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧(abs ∘ − )𝐵) < 𝑦 ↔ (abs‘(𝑧𝐵)) < 𝑦))
9287, 91bitrd 279 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (abs‘(𝑧𝐵)) < 𝑦))
93 simplrl 774 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ+)
9493rpxrd 13024 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ*)
95 simpllr 773 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐶 ∈ ℂ)
96 eldifi 4126 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐴)
97 ffvelcdm 7083 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
9870, 96, 97syl2an 595 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝐹𝑧) ∈ ℂ)
99 elbl3 24217 . . . . . . . . . . . . 13 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℝ*) ∧ (𝐶 ∈ ℂ ∧ (𝐹𝑧) ∈ ℂ)) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥))
10079, 94, 95, 98, 99syl22anc 836 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥))
10188cnmetdval 24606 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
10298, 95, 101syl2anc 583 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
103102breq1d 5158 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥 ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
104100, 103bitrd 279 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
10592, 104imbi12d 344 . . . . . . . . . 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 3174 . . . . . . . . 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 3964 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})))
108107biancomi 462 . . . . . . . . . . . 12 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
109108imbi1i 349 . . . . . . . . . . 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 450 . . . . . . . . . . 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 276 . . . . . . . . . 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 3088 . . . . . . . . 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 450 . . . . . . . . . . 11 (((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
114 eldifsn 4790 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧𝐴𝑧𝐵))
115114imbi1i 349 . . . . . . . . . . 11 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ ((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
116 impexp 450 . . . . . . . . . . . 12 (((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
117116imbi2i 336 . . . . . . . . . . 11 ((𝑧𝐴 → ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
118113, 115, 1173bitr4i 303 . . . . . . . . . 10 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
119118ralbii2 3088 . . . . . . . . 9 (∀𝑧 ∈ (𝐴 ∖ {𝐵})((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
120106, 112, 1193bitr3g 313 . . . . . . . 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 279 . . . . . . 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 467 . . . . . 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 3175 . . . . 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 3174 . . . 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 279 . . 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 578 . 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 279 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 205  wa 395   = wceq 1540  wcel 2105  wne 2939  wral 3060  wrex 3069  cdif 3945  cin 3947  wss 3948  {csn 4628   class class class wbr 5148  dom cdm 5676  cima 5679  ccom 5680  Fun wfun 6537  wf 6539  cfv 6543  (class class class)co 7412  cc 11114  *cxr 11254   < clt 11255  cmin 11451  +crp 12981  abscabs 15188  TopOpenctopn 17374  ∞Metcxmet 21217  ballcbl 21219  fldccnfld 21232   lim climc 25710
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fi 9412  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-q 12940  df-rp 12982  df-xneg 13099  df-xadd 13100  df-xmul 13101  df-fz 13492  df-seq 13974  df-exp 14035  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-struct 17087  df-slot 17122  df-ndx 17134  df-base 17152  df-plusg 17217  df-mulr 17218  df-starv 17219  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-rest 17375  df-topn 17376  df-topgen 17396  df-psmet 21224  df-xmet 21225  df-met 21226  df-bl 21227  df-mopn 21228  df-cnfld 21233  df-top 22715  df-topon 22732  df-topsp 22754  df-bases 22768  df-cnp 23051  df-xms 24145  df-ms 24146  df-limc 25714
This theorem is referenced by:  dveflem  25830  dvferm1  25836  dvferm2  25838  lhop1  25866  ftc1lem6  25895  ulmdvlem3  26252  unblimceq0  35846  ftc1cnnc  37023  mullimc  44790  ellimcabssub0  44791  limcdm0  44792  mullimcf  44797  constlimc  44798  idlimc  44800  limcperiod  44802  limcrecl  44803  limcleqr  44818  neglimc  44821  addlimc  44822  0ellimcdiv  44823  limclner  44825  fperdvper  45093  ioodvbdlimc1lem2  45106  ioodvbdlimc2lem  45108
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