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Theorem cnheibor 22967
Description: Heine-Borel theorem for complex numbers. A subset of is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
cnheibor.2 𝐽 = (TopOpen‘ℂfld)
cnheibor.3 𝑇 = (𝐽t 𝑋)
Assertion
Ref Expression
cnheibor (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)))
Distinct variable groups:   𝑥,𝑟,𝑇   𝐽,𝑟,𝑥   𝑋,𝑟,𝑥

Proof of Theorem cnheibor
Dummy variables 𝑧 𝑢 𝑓 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnheibor.2 . . . . . 6 𝐽 = (TopOpen‘ℂfld)
21cnfldhaus 22801 . . . . 5 𝐽 ∈ Haus
32a1i 11 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝐽 ∈ Haus)
4 simpl 468 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
5 cnheibor.3 . . . . 5 𝑇 = (𝐽t 𝑋)
6 simpr 471 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
75, 6syl5eqelr 2855 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽t 𝑋) ∈ Comp)
81cnfldtopon 22799 . . . . . 6 𝐽 ∈ (TopOn‘ℂ)
98toponunii 20934 . . . . 5 ℂ = 𝐽
109hauscmp 21424 . . . 4 ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
113, 4, 7, 10syl3anc 1476 . . 3 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
121cnfldtop 22800 . . . . . . . . . . 11 𝐽 ∈ Top
139restuni 21180 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = (𝐽t 𝑋))
1412, 4, 13sylancr 575 . . . . . . . . . 10 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = (𝐽t 𝑋))
155unieqi 4583 . . . . . . . . . 10 𝑇 = (𝐽t 𝑋)
1614, 15syl6eqr 2823 . . . . . . . . 9 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = 𝑇)
1716eleq2d 2836 . . . . . . . 8 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥𝑋𝑥 𝑇))
1817biimpar 463 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → 𝑥𝑋)
1912a1i 11 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝐽 ∈ Top)
20 cnex 10217 . . . . . . . . . . . 12 ℂ ∈ V
21 ssexg 4938 . . . . . . . . . . . 12 ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
224, 20, 21sylancl 574 . . . . . . . . . . 11 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
2322adantr 466 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑋 ∈ V)
24 cnxmet 22789 . . . . . . . . . . . 12 (abs ∘ − ) ∈ (∞Met‘ℂ)
2524a1i 11 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs ∘ − ) ∈ (∞Met‘ℂ))
26 0cnd 10233 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ∈ ℂ)
274sselda 3752 . . . . . . . . . . . . . 14 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ℂ)
2827abscld 14376 . . . . . . . . . . . . 13 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) ∈ ℝ)
29 peano2re 10409 . . . . . . . . . . . . 13 ((abs‘𝑥) ∈ ℝ → ((abs‘𝑥) + 1) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . . 12 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ)
3130rexrd 10289 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ*)
321cnfldtopn 22798 . . . . . . . . . . . 12 𝐽 = (MetOpen‘(abs ∘ − ))
3332blopn 22518 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
3425, 26, 31, 33syl3anc 1476 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
35 elrestr 16290 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 ∈ V ∧ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽t 𝑋))
3619, 23, 34, 35syl3anc 1476 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽t 𝑋))
3736, 5syl6eleqr 2861 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
38 0cn 10232 . . . . . . . . . . . . . 14 0 ∈ ℂ
39 eqid 2771 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
4039cnmetdval 22787 . . . . . . . . . . . . . 14 ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
4138, 40mpan 670 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
42 df-neg 10469 . . . . . . . . . . . . . . 15 -𝑥 = (0 − 𝑥)
4342fveq2i 6333 . . . . . . . . . . . . . 14 (abs‘-𝑥) = (abs‘(0 − 𝑥))
44 absneg 14218 . . . . . . . . . . . . . 14 (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
4543, 44syl5eqr 2819 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
4641, 45eqtrd 2805 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4727, 46syl 17 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4828ltp1d 11154 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
4947, 48eqbrtrd 4808 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
50 elbl 22406 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ*) → (𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))))
5125, 26, 31, 50syl3anc 1476 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))))
5227, 49, 51mpbir2and 692 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
53 simpr 471 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥𝑋)
5452, 53elind 3949 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
5527absge0d 14384 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ≤ (abs‘𝑥))
5628, 55ge0p1rpd 12098 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ+)
57 eqid 2771 . . . . . . . . 9 ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
58 oveq2 6799 . . . . . . . . . . . 12 (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
5958ineq1d 3964 . . . . . . . . . . 11 (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
6059eqeq2d 2781 . . . . . . . . . 10 (𝑟 = ((abs‘𝑥) + 1) → (((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
6160rspcev 3460 . . . . . . . . 9 ((((abs‘𝑥) + 1) ∈ ℝ+ ∧ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)) → ∃𝑟 ∈ ℝ+ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
6256, 57, 61sylancl 574 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ∃𝑟 ∈ ℝ+ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
63 eleq2 2839 . . . . . . . . . 10 (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥𝑢𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
64 eqeq1 2775 . . . . . . . . . . 11 (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6564rexbidv 3200 . . . . . . . . . 10 (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ ∃𝑟 ∈ ℝ+ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6663, 65anbi12d 616 . . . . . . . . 9 (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → ((𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)) ↔ (𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∧ ∃𝑟 ∈ ℝ+ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))))
6766rspcev 3460 . . . . . . . 8 ((((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇 ∧ (𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∧ ∃𝑟 ∈ ℝ+ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6837, 54, 62, 67syl12anc 1474 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6918, 68syldan 579 . . . . . 6 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
7069ralrimiva 3115 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 𝑇𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
71 eqid 2771 . . . . . 6 𝑇 = 𝑇
72 oveq2 6799 . . . . . . . 8 (𝑟 = (𝑓𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓𝑢)))
7372ineq1d 3964 . . . . . . 7 (𝑟 = (𝑓𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))
7473eqeq2d 2781 . . . . . 6 (𝑟 = (𝑓𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋)))
7571, 74cmpcovf 21408 . . . . 5 ((𝑇 ∈ Comp ∧ ∀𝑥 𝑇𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))) → ∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)( 𝑇 = 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))))
766, 70, 75syl2anc 573 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)( 𝑇 = 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))))
7716ad4antr 712 . . . . . . . . . . . . . 14 ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) → 𝑋 = 𝑇)
78 simpllr 760 . . . . . . . . . . . . . 14 ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) → 𝑇 = 𝑠)
7977, 78eqtrd 2805 . . . . . . . . . . . . 13 ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) → 𝑋 = 𝑠)
8079eleq2d 2836 . . . . . . . . . . . 12 ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) → (𝑥𝑋𝑥 𝑠))
81 eluni2 4578 . . . . . . . . . . . 12 (𝑥 𝑠 ↔ ∃𝑧𝑠 𝑥𝑧)
8280, 81syl6bb 276 . . . . . . . . . . 11 ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) → (𝑥𝑋 ↔ ∃𝑧𝑠 𝑥𝑧))
83 elssuni 4603 . . . . . . . . . . . . . . . . . 18 (𝑧𝑠𝑧 𝑠)
8483ad2antrl 707 . . . . . . . . . . . . . . . . 17 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑧 𝑠)
8579adantr 466 . . . . . . . . . . . . . . . . 17 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑋 = 𝑠)
8684, 85sseqtr4d 3791 . . . . . . . . . . . . . . . 16 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑧𝑋)
87 simp-6l 776 . . . . . . . . . . . . . . . 16 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑋 ⊆ ℂ)
8886, 87sstrd 3762 . . . . . . . . . . . . . . 15 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑧 ⊆ ℂ)
89 simprr 756 . . . . . . . . . . . . . . 15 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑥𝑧)
9088, 89sseldd 3753 . . . . . . . . . . . . . 14 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑥 ∈ ℂ)
9190abscld 14376 . . . . . . . . . . . . 13 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (abs‘𝑥) ∈ ℝ)
92 simplrl 762 . . . . . . . . . . . . 13 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑟 ∈ ℝ)
93 simprl 754 . . . . . . . . . . . . . . . . 17 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → 𝑓:𝑠⟶ℝ+)
9493ad2antrr 705 . . . . . . . . . . . . . . . 16 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑓:𝑠⟶ℝ+)
95 simprl 754 . . . . . . . . . . . . . . . 16 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑧𝑠)
9694, 95ffvelrnd 6501 . . . . . . . . . . . . . . 15 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (𝑓𝑧) ∈ ℝ+)
9796rpred 12068 . . . . . . . . . . . . . 14 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (𝑓𝑧) ∈ ℝ)
9890, 46syl 17 . . . . . . . . . . . . . . 15 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
99 inss1 3981 . . . . . . . . . . . . . . . . . 18 ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋) ⊆ (0(ball‘(abs ∘ − ))(𝑓𝑧))
100 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧𝑢 = 𝑧)
101 fveq2 6330 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
102101oveq2d 6807 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓𝑢)) = (0(ball‘(abs ∘ − ))(𝑓𝑧)))
103102ineq1d 3964 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋))
104100, 103eqeq12d 2786 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋)))
105 simprr 756 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))
106105ad2antrr 705 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))
107104, 106, 95rspcdva 3466 . . . . . . . . . . . . . . . . . . 19 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋))
10889, 107eleqtrd 2852 . . . . . . . . . . . . . . . . . 18 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋))
10999, 108sseldi 3750 . . . . . . . . . . . . . . . . 17 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓𝑧)))
11024a1i 11 . . . . . . . . . . . . . . . . . 18 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
111 0cnd 10233 . . . . . . . . . . . . . . . . . 18 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → 0 ∈ ℂ)
11296rpxrd 12069 . . . . . . . . . . . . . . . . . 18 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (𝑓𝑧) ∈ ℝ*)
113 elbl 22406 . . . . . . . . . . . . . . . . . 18 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (𝑓𝑧) ∈ ℝ*) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓𝑧)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓𝑧))))
114110, 111, 112, 113syl3anc 1476 . . . . . . . . . . . . . . . . 17 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓𝑧)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓𝑧))))
115109, 114mpbid 222 . . . . . . . . . . . . . . . 16 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓𝑧)))
116115simprd 483 . . . . . . . . . . . . . . 15 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (0(abs ∘ − )𝑥) < (𝑓𝑧))
11798, 116eqbrtrrd 4810 . . . . . . . . . . . . . 14 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (abs‘𝑥) < (𝑓𝑧))
118101breq1d 4796 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → ((𝑓𝑢) ≤ 𝑟 ↔ (𝑓𝑧) ≤ 𝑟))
119 simplrr 763 . . . . . . . . . . . . . . 15 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)
120118, 119, 95rspcdva 3466 . . . . . . . . . . . . . 14 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (𝑓𝑧) ≤ 𝑟)
12191, 97, 92, 117, 120ltletrd 10397 . . . . . . . . . . . . 13 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (abs‘𝑥) < 𝑟)
12291, 92, 121ltled 10385 . . . . . . . . . . . 12 (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) ∧ (𝑧𝑠𝑥𝑧)) → (abs‘𝑥) ≤ 𝑟)
123122rexlimdvaa 3180 . . . . . . . . . . 11 ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) → (∃𝑧𝑠 𝑥𝑧 → (abs‘𝑥) ≤ 𝑟))
12482, 123sylbid 230 . . . . . . . . . 10 ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) → (𝑥𝑋 → (abs‘𝑥) ≤ 𝑟))
125124ralrimiv 3114 . . . . . . . . 9 ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)) → ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)
126 inss2 3982 . . . . . . . . . . 11 (𝒫 𝑇 ∩ Fin) ⊆ Fin
127 simpllr 760 . . . . . . . . . . 11 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → 𝑠 ∈ (𝒫 𝑇 ∩ Fin))
128126, 127sseldi 3750 . . . . . . . . . 10 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → 𝑠 ∈ Fin)
129 ffvelrn 6498 . . . . . . . . . . . . 13 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ+)
130129rpred 12068 . . . . . . . . . . . 12 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ)
131130ralrimiva 3115 . . . . . . . . . . 11 (𝑓:𝑠⟶ℝ+ → ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ)
132131ad2antrl 707 . . . . . . . . . 10 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ)
133 fimaxre3 11170 . . . . . . . . . 10 ((𝑠 ∈ Fin ∧ ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)
134128, 132, 133syl2anc 573 . . . . . . . . 9 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → ∃𝑟 ∈ ℝ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)
135125, 134reximddv 3166 . . . . . . . 8 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)
136135ex 397 . . . . . . 7 ((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) → ((𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋)) → ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
137136exlimdv 2013 . . . . . 6 ((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) → (∃𝑓(𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋)) → ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
138137expimpd 441 . . . . 5 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) → (( 𝑇 = 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
139138rexlimdva 3179 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)( 𝑇 = 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
14076, 139mpd 15 . . 3 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)
14111, 140jca 501 . 2 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
142 eqid 2771 . . . . . 6 (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
143 eqid 2771 . . . . . 6 ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) “ ((-𝑟[,]𝑟) × (-𝑟[,]𝑟))) = ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) “ ((-𝑟[,]𝑟) × (-𝑟[,]𝑟)))
1441, 5, 142, 143cnheiborlem 22966 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
145144rexlimdvaa 3180 . . . 4 (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟𝑇 ∈ Comp))
146145imp 393 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
147146adantl 467 . 2 ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
148141, 147impbida 802 1 (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  cin 3722  wss 3723  𝒫 cpw 4297   cuni 4574   class class class wbr 4786   × cxp 5247  cima 5252  ccom 5253  wf 6025  cfv 6029  (class class class)co 6791  cmpt2 6793  Fincfn 8107  cc 10134  cr 10135  0cc0 10136  1c1 10137  ici 10138   + caddc 10139   · cmul 10141  *cxr 10273   < clt 10274  cle 10275  cmin 10466  -cneg 10467  +crp 12028  [,]cicc 12376  abscabs 14175  t crest 16282  TopOpenctopn 16283  ∞Metcxmt 19939  ballcbl 19941  fldccnfld 19954  Topctop 20911  Clsdccld 21034  Hauscha 21326  Compccmp 21403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-inf2 8700  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-pre-sup 10214  ax-addf 10215  ax-mulf 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-isom 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-of 7042  df-om 7211  df-1st 7313  df-2nd 7314  df-supp 7445  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-er 7894  df-map 8009  df-ixp 8061  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-fsupp 8430  df-fi 8471  df-sup 8502  df-inf 8503  df-oi 8569  df-card 8963  df-cda 9190  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-2 11279  df-3 11280  df-4 11281  df-5 11282  df-6 11283  df-7 11284  df-8 11285  df-9 11286  df-n0 11493  df-z 11578  df-dec 11694  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12144  df-xadd 12145  df-xmul 12146  df-ioo 12377  df-icc 12380  df-fz 12527  df-fzo 12667  df-seq 13002  df-exp 13061  df-hash 13315  df-cj 14040  df-re 14041  df-im 14042  df-sqrt 14176  df-abs 14177  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16155  df-mulr 16156  df-starv 16157  df-sca 16158  df-vsca 16159  df-ip 16160  df-tset 16161  df-ple 16162  df-ds 16165  df-unif 16166  df-hom 16167  df-cco 16168  df-rest 16284  df-topn 16285  df-0g 16303  df-gsum 16304  df-topgen 16305  df-pt 16306  df-prds 16309  df-xrs 16363  df-qtop 16368  df-imas 16369  df-xps 16371  df-mre 16447  df-mrc 16448  df-acs 16450  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-cnfld 19955  df-top 20912  df-topon 20929  df-topsp 20951  df-bases 20964  df-cld 21037  df-cls 21039  df-cn 21245  df-cnp 21246  df-haus 21333  df-cmp 21404  df-tx 21579  df-hmeo 21772  df-xms 22338  df-ms 22339  df-tms 22340  df-cncf 22894
This theorem is referenced by:  cnllycmp  22968  cncmet  23331  ftalem3  25015
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