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

Theorem ellimc3 24049
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 2825 . . 3 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
51, 2, 3, 4ellimc2 24047 . 2 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
6 cnxmet 22953 . . . . . . . . . 10 (abs ∘ − ) ∈ (∞Met‘ℂ)
76a1i 11 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (abs ∘ − ) ∈ (∞Met‘ℂ))
8 simplr 785 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ ℂ)
9 simpr 479 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
10 blcntr 22595 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
117, 8, 9, 10syl3anc 1494 . . . . . . . 8 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥))
12 rpxr 12130 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
1312adantl 475 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ*)
144cnfldtopn 22962 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
1514blopn 22682 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ*) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂfld))
167, 8, 13, 15syl3anc 1494 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) → (𝐶(ball‘(abs ∘ − ))𝑥) ∈ (TopOpen‘ℂfld))
17 eleq2 2895 . . . . . . . . . . 11 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → (𝐶𝑢𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑥)))
18 sseq2 3852 . . . . . . . . . . . . 13 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)))
1918anbi2d 622 . . . . . . . . . . . 12 (𝑢 = (𝐶(ball‘(abs ∘ − ))𝑥) → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥))))
2019rexbidv 3262 . . . . . . . . . . 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 336 . . . . . . . . . 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 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)))))
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 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))))
2514mopni2 22675 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ∃𝑦 ∈ ℝ+ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
266, 25mp3an1 1576 . . . . . . . . . 10 ((𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐵𝑣) → ∃𝑦 ∈ ℝ+ (𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣)
27 ssrin 4064 . . . . . . . . . . . . 13 ((𝐵(ball‘(abs ∘ − ))𝑦) ⊆ 𝑣 → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})))
28 imass2 5746 . . . . . . . . . . . . 13 (((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝑣 ∩ (𝐴 ∖ {𝐵})) → (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))))
29 sstr2 3834 . . . . . . . . . . . . 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 3224 . . . . . . . . . 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 448 . . . . . . . 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 3237 . . . . . . 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 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)))
3736ralrimdva 3178 . . . . 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 22675 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → ∃𝑥 ∈ ℝ+ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
396, 38mp3an1 1576 . . . . . . . . 9 ((𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝐶𝑢) → ∃𝑥 ∈ ℝ+ (𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢)
40 r19.29r 3283 . . . . . . . . . . 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 721 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ ℂ)
43 simpr 479 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
4443rpxrd 12164 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ*)
4514blopn 22682 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ*) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld))
4641, 42, 44, 45syl3anc 1494 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝐵(ball‘(abs ∘ − ))𝑦) ∈ (TopOpen‘ℂfld))
47 blcntr 22595 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
4841, 42, 43, 47syl3anc 1494 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
49 eleq2 2895 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐵𝑣𝐵 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
50 ineq1 4036 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})))
5150imaeq2d 5711 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝐵(ball‘(abs ∘ − ))𝑦) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵}))))
5251sseq1d 3857 . . . . . . . . . . . . . . . . . . 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 624 . . . . . . . . . . . . . . . . . 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 3526 . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . 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 3240 . . . . . . . . . . . . . 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 3834 . . . . . . . . . . . . . . . . 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 605 . . . . . . . . . . . . . . 15 ((𝐶(ball‘(abs ∘ − ))𝑥) ⊆ 𝑢 → ((𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ (𝐶(ball‘(abs ∘ − ))𝑥)) → (𝐵𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
6160reximdv 3224 . . . . . . . . . . . . . 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 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))))
6362impd 400 . . . . . . . . . . . 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 3240 . . . . . . . . . . 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 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)))
6665expd 406 . . . . . . . . 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 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))))
6867expdimp 446 . . . . . . 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 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))))
7069ralrimdva 3178 . . . . 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 204 . . . 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 717 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → 𝐹:𝐴⟶ℂ)
7372ffund 6286 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → Fun 𝐹)
74 inss2 4060 . . . . . . . . . 10 ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
75 difss 3966 . . . . . . . . . . 11 (𝐴 ∖ {𝐵}) ⊆ 𝐴
7672fdmd 6291 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → dom 𝐹 = 𝐴)
7775, 76syl5sseqr 3879 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ dom 𝐹)
7874, 77syl5ss 3838 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
79 funimass4 6498 . . . . . . . . 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)))
8073, 78, 79syl2anc 579 . . . . . . . 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)))
816a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (abs ∘ − ) ∈ (∞Met‘ℂ))
82 simplrr 796 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ+)
8382rpxrd 12164 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℝ*)
843ad3antrrr 721 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
8575, 2syl5ss 3838 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8685ad2antrr 717 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
8786sselda 3827 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑧 ∈ ℂ)
88 elbl3 22574 . . . . . . . . . . . . 13 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℝ*) ∧ (𝐵 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
8981, 83, 84, 87, 88syl22anc 872 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧(abs ∘ − )𝐵) < 𝑦))
90 eqid 2825 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
9190cnmetdval 22951 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9287, 84, 91syl2anc 579 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧(abs ∘ − )𝐵) = (abs‘(𝑧𝐵)))
9392breq1d 4885 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝑧(abs ∘ − )𝐵) < 𝑦 ↔ (abs‘(𝑧𝐵)) < 𝑦))
9489, 93bitrd 271 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (abs‘(𝑧𝐵)) < 𝑦))
95 simplrl 795 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ+)
9695rpxrd 12164 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ*)
97 simpllr 793 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → 𝐶 ∈ ℂ)
98 eldifi 3961 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐴)
99 ffvelrn 6611 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
10072, 98, 99syl2an 589 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (𝐹𝑧) ∈ ℂ)
101 elbl3 22574 . . . . . . . . . . . . 13 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℝ*) ∧ (𝐶 ∈ ℂ ∧ (𝐹𝑧) ∈ ℂ)) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥))
10281, 96, 97, 100, 101syl22anc 872 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ ((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥))
10390cnmetdval 22951 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
104100, 97, 103syl2anc 579 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧)(abs ∘ − )𝐶) = (abs‘((𝐹𝑧) − 𝐶)))
105104breq1d 4885 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (((𝐹𝑧)(abs ∘ − )𝐶) < 𝑥 ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
106102, 105bitrd 271 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℝ+𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → ((𝐹𝑧) ∈ (𝐶(ball‘(abs ∘ − ))𝑥) ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
10794, 106imbi12d 336 . . . . . . . . . 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)))
108107ralbidva 3194 . . . . . . . . 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)))
109 elin 4025 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})))
110 ancom 454 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
111109, 110bitri 267 . . . . . . . . . . . 12 (𝑧 ∈ ((𝐵(ball‘(abs ∘ − ))𝑦) ∩ (𝐴 ∖ {𝐵})) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
112111imbi1i 341 . . . . . . . . . . 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)))
113 impexp 443 . . . . . . . . . . 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))))
114112, 113bitr2i 268 . . . . . . . . . 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)))
115114ralbii2 3187 . . . . . . . . 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))
116 impexp 443 . . . . . . . . . . 11 (((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
117 eldifsn 4538 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧𝐴𝑧𝐵))
118117imbi1i 341 . . . . . . . . . . 11 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ ((𝑧𝐴𝑧𝐵) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
119 impexp 443 . . . . . . . . . . . 12 (((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
120119imbi2i 328 . . . . . . . . . . 11 ((𝑧𝐴 → ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → (𝑧𝐵 → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
121116, 118, 1203bitr4i 295 . . . . . . . . . 10 ((𝑧 ∈ (𝐴 ∖ {𝐵}) → ((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝑧𝐴 → ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
122121ralbii2 3187 . . . . . . . . 9 (∀𝑧 ∈ (𝐴 ∖ {𝐵})((abs‘(𝑧𝐵)) < 𝑦 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
123108, 115, 1223bitr3g 305 . . . . . . . 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)))
12480, 123bitrd 271 . . . . . . 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)))
125124anassrs 461 . . . . . 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)))
126125rexbidva 3259 . . . . 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)))
127126ralbidva 3194 . . . 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)))
12871, 127bitrd 271 . . 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)))
129128pm5.32da 574 . 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))))
1305, 129bitrd 271 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 198  wa 386   = wceq 1656  wcel 2164  wne 2999  wral 3117  wrex 3118  cdif 3795  cin 3797  wss 3798  {csn 4399   class class class wbr 4875  dom cdm 5346  cima 5349  ccom 5350  Fun wfun 6121  wf 6123  cfv 6127  (class class class)co 6910  cc 10257  *cxr 10397   < clt 10398  cmin 10592  +crp 12119  abscabs 14358  TopOpenctopn 16442  ∞Metcxmet 20098  ballcbl 20100  fldccnfld 20113   lim climc 24032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-fi 8592  df-sup 8623  df-inf 8624  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-7 11426  df-8 11427  df-9 11428  df-n0 11626  df-z 11712  df-dec 11829  df-uz 11976  df-q 12079  df-rp 12120  df-xneg 12239  df-xadd 12240  df-xmul 12241  df-fz 12627  df-seq 13103  df-exp 13162  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-struct 16231  df-ndx 16232  df-slot 16233  df-base 16235  df-plusg 16325  df-mulr 16326  df-starv 16327  df-tset 16331  df-ple 16332  df-ds 16334  df-unif 16335  df-rest 16443  df-topn 16444  df-topgen 16464  df-psmet 20105  df-xmet 20106  df-met 20107  df-bl 20108  df-mopn 20109  df-cnfld 20114  df-top 21076  df-topon 21093  df-topsp 21115  df-bases 21128  df-cnp 21410  df-xms 22502  df-ms 22503  df-limc 24036
This theorem is referenced by:  dveflem  24148  dvferm1  24154  dvferm2  24156  lhop1  24183  ftc1lem6  24210  ulmdvlem3  24562  unblimceq0  33025  ftc1cnnc  34022  mullimc  40637  ellimcabssub0  40638  limcdm0  40639  mullimcf  40644  constlimc  40645  idlimc  40647  limcperiod  40649  limcrecl  40650  limcleqr  40665  neglimc  40668  addlimc  40669  0ellimcdiv  40670  limclner  40672  fperdvper  40922  ioodvbdlimc1lem2  40936  ioodvbdlimc2lem  40938
  Copyright terms: Public domain W3C validator