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Theorem ellimc3 25043
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 25041 . 2 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
6 cnxmet 23936 . . . . . . . . 9 (abs ∘ − ) ∈ (∞Met‘ℂ)
7 simplr 766 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ ℂ)
8 simpr 485 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
9 blcntr 23566 . . . . . . . . 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 12739 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
1211adantl 482 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ*)
134cnfldtopn 23945 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
1413blopn 23656 . . . . . . . . . 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 2827 . . . . . . . . . . 11 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶𝑢𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
17 sseq2 3947 . . . . . . . . . . . . 13 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
1817anbi2d 629 . . . . . . . . . . . 12 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
1918rexbidv 3226 . . . . . . . . . . 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 345 . . . . . . . . . 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 3557 . . . . . . . . 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 23649 . . . . . . . . . . 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 4167 . . . . . . . . . . . . 13 ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
27 imass2 6010 . . . . . . . . . . . . 13 (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
28 sstr2 3928 . . . . . . . . . . . . 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 3202 . . . . . . . . . 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 455 . . . . . . . 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 3210 . . . . . . 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 3106 . . . . 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 23649 . . . . . . . . . 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 3185 . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
4241rpxrd 12773 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ*)
4313blopn 23656 . . . . . . . . . . . . . . . . 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 23566 . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵𝑣𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
48 ineq1 4139 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
4948imaeq2d 5969 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
5049sseq1d 3952 . . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . . 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 3561 . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 3213 . . . . . . . . . . . . . 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 3928 . . . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . 15 ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
5958reximdv 3202 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 3213 . . . . . . . . . . 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 416 . . . . . . . . 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 453 . . . . . . 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 3106 . . . . 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 6604 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → Fun 𝐹)
72 inss2 4163 . . . . . . . . . 10 ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
73 difss 4066 . . . . . . . . . . 11 (𝐴 ∖ {𝐵}) ⊆ 𝐴
7470fdmd 6611 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → dom 𝐹 = 𝐴)
7573, 74sseqtrrid 3974 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
7672, 75sstrid 3932 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
77 funimass4 6834 . . . . . . . . 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 584 . . . . . . . 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 12773 . . . . . . . . . . . . 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 3932 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8483ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8584sselda 3921 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
86 elbl3 23545 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
8988cnmetdval 23934 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9085, 82, 89syl2anc 584 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9190breq1d 5084 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧(abs ∘ − )𝐵) < 𝑦 ↔ (abs‘(𝑧𝐵)) < 𝑦))
9287, 91bitrd 278 . . . . . . . . . . 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 12773 . . . . . . . . . . . . 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 4061 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐴)
97 ffvelrn 6959 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
9870, 96, 97syl2an 596 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝐹𝑧) ∈ ℂ)
99 elbl3 23545 . . . . . . . . . . . . 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 23934 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
10298, 95, 101syl2anc 584 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
103102breq1d 5084 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥 ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
104100, 103bitrd 278 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
10592, 104imbi12d 345 . . . . . . . . . 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 3111 . . . . . . . . 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 3903 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})))
108107biancomi 463 . . . . . . . . . . . 12 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
109108imbi1i 350 . . . . . . . . . . 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 451 . . . . . . . . . . 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 275 . . . . . . . . . 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 3090 . . . . . . . . 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 451 . . . . . . . . . . 11 (((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
114 eldifsn 4720 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧𝐴𝑧𝐵))
115114imbi1i 350 . . . . . . . . . . 11 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ ((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
116 impexp 451 . . . . . . . . . . . 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 3090 . . . . . . . . 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 278 . . . . . . 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 468 . . . . . 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 3225 . . . . 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 3111 . . . 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 278 . . 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 579 . 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 278 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 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  wrex 3065  cdif 3884  cin 3886  wss 3887  {csn 4561   class class class wbr 5074  dom cdm 5589  cima 5592  ccom 5593  Fun wfun 6427  wf 6429  cfv 6433  (class class class)co 7275  cc 10869  *cxr 11008   < clt 11009  cmin 11205  +crp 12730  abscabs 14945  TopOpenctopn 17132  ∞Metcxmet 20582  ballcbl 20584  fldccnfld 20597   lim climc 25026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-fz 13240  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-struct 16848  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-mulr 16976  df-starv 16977  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-rest 17133  df-topn 17134  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cnp 22379  df-xms 23473  df-ms 23474  df-limc 25030
This theorem is referenced by:  dveflem  25143  dvferm1  25149  dvferm2  25151  lhop1  25178  ftc1lem6  25205  ulmdvlem3  25561  unblimceq0  34687  ftc1cnnc  35849  mullimc  43157  ellimcabssub0  43158  limcdm0  43159  mullimcf  43164  constlimc  43165  idlimc  43167  limcperiod  43169  limcrecl  43170  limcleqr  43185  neglimc  43188  addlimc  43189  0ellimcdiv  43190  limclner  43192  fperdvper  43460  ioodvbdlimc1lem2  43473  ioodvbdlimc2lem  43475
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