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Theorem cnheibor 24925
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 . . . . 5 𝐽 = (TopOpen‘ℂfld)
21cnfldhaus 24745 . . . 4 𝐽 ∈ Haus
3 simpl 481 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
4 cnheibor.3 . . . . 5 𝑇 = (𝐽t 𝑋)
5 simpr 483 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
64, 5eqeltrrid 2830 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽t 𝑋) ∈ Comp)
71cnfldtopon 24743 . . . . . 6 𝐽 ∈ (TopOn‘ℂ)
87toponunii 22862 . . . . 5 ℂ = 𝐽
98hauscmp 23355 . . . 4 ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
102, 3, 6, 9mp3an2i 1462 . . 3 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
111cnfldtop 24744 . . . . . . . . . . 11 𝐽 ∈ Top
128restuni 23110 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = (𝐽t 𝑋))
1311, 3, 12sylancr 585 . . . . . . . . . 10 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = (𝐽t 𝑋))
144unieqi 4921 . . . . . . . . . 10 𝑇 = (𝐽t 𝑋)
1513, 14eqtr4di 2783 . . . . . . . . 9 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = 𝑇)
1615eleq2d 2811 . . . . . . . 8 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥𝑋𝑥 𝑇))
1716biimpar 476 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → 𝑥𝑋)
18 cnex 11221 . . . . . . . . . . . 12 ℂ ∈ V
19 ssexg 5324 . . . . . . . . . . . 12 ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
203, 18, 19sylancl 584 . . . . . . . . . . 11 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
2120adantr 479 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑋 ∈ V)
22 cnxmet 24733 . . . . . . . . . . 11 (abs ∘ − ) ∈ (∞Met‘ℂ)
23 0cnd 11239 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ∈ ℂ)
243sselda 3976 . . . . . . . . . . . . . 14 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ℂ)
2524abscld 15419 . . . . . . . . . . . . 13 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) ∈ ℝ)
26 peano2re 11419 . . . . . . . . . . . . 13 ((abs‘𝑥) ∈ ℝ → ((abs‘𝑥) + 1) ∈ ℝ)
2725, 26syl 17 . . . . . . . . . . . 12 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ)
2827rexrd 11296 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ*)
291cnfldtopn 24742 . . . . . . . . . . . 12 𝐽 = (MetOpen‘(abs ∘ − ))
3029blopn 24453 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
3122, 23, 28, 30mp3an2i 1462 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
32 elrestr 17413 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 ∈ V ∧ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽t 𝑋))
3311, 21, 31, 32mp3an2i 1462 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽t 𝑋))
3433, 4eleqtrrdi 2836 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
35 0cn 11238 . . . . . . . . . . . . . 14 0 ∈ ℂ
36 eqid 2725 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
3736cnmetdval 24731 . . . . . . . . . . . . . 14 ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
3835, 37mpan 688 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
39 df-neg 11479 . . . . . . . . . . . . . . 15 -𝑥 = (0 − 𝑥)
4039fveq2i 6899 . . . . . . . . . . . . . 14 (abs‘-𝑥) = (abs‘(0 − 𝑥))
41 absneg 15260 . . . . . . . . . . . . . 14 (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
4240, 41eqtr3id 2779 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
4338, 42eqtrd 2765 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4424, 43syl 17 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4525ltp1d 12177 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
4644, 45eqbrtrd 5171 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
47 elbl 24338 . . . . . . . . . . 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))))
4822, 23, 28, 47mp3an2i 1462 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))))
4924, 46, 48mpbir2and 711 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
50 simpr 483 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥𝑋)
5149, 50elind 4192 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
5224absge0d 15427 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ≤ (abs‘𝑥))
5325, 52ge0p1rpd 13081 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ+)
54 eqid 2725 . . . . . . . . 9 ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
55 oveq2 7427 . . . . . . . . . . 11 (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
5655ineq1d 4209 . . . . . . . . . 10 (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
5756rspceeqv 3628 . . . . . . . . 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))
5853, 54, 57sylancl 584 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ∃𝑟 ∈ ℝ+ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
59 eleq2 2814 . . . . . . . . . 10 (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥𝑢𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
60 eqeq1 2729 . . . . . . . . . . 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)))
6160rexbidv 3168 . . . . . . . . . 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)))
6259, 61anbi12d 630 . . . . . . . . 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))))
6362rspcev 3606 . . . . . . . 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)))
6434, 51, 58, 63syl12anc 835 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6517, 64syldan 589 . . . . . 6 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6665ralrimiva 3135 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 𝑇𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
67 eqid 2725 . . . . . 6 𝑇 = 𝑇
68 oveq2 7427 . . . . . . . 8 (𝑟 = (𝑓𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓𝑢)))
6968ineq1d 4209 . . . . . . 7 (𝑟 = (𝑓𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))
7069eqeq2d 2736 . . . . . 6 (𝑟 = (𝑓𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋)))
7167, 70cmpcovf 23339 . . . . 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))))
725, 66, 71syl2anc 582 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)( 𝑇 = 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))))
7315ad4antr 730 . . . . . . . . . . . . . 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)
74 simpllr 774 . . . . . . . . . . . . . 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)
7573, 74eqtrd 2765 . . . . . . . . . . . . 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)
7675eleq2d 2811 . . . . . . . . . . . 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))
77 eluni2 4913 . . . . . . . . . . . 12 (𝑥 𝑠 ↔ ∃𝑧𝑠 𝑥𝑧)
7876, 77bitrdi 286 . . . . . . . . . . 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))
79 elssuni 4941 . . . . . . . . . . . . . . . . . 18 (𝑧𝑠𝑧 𝑠)
8079ad2antrl 726 . . . . . . . . . . . . . . . . 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)
8175adantr 479 . . . . . . . . . . . . . . . . 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)
8280, 81sseqtrrd 4018 . . . . . . . . . . . . . . . 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)
83 simp-6l 785 . . . . . . . . . . . . . . . 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)
8482, 83sstrd 3987 . . . . . . . . . . . . . . 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)
85 simprr 771 . . . . . . . . . . . . . . 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)
8684, 85sseldd 3977 . . . . . . . . . . . . . 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)
8786abscld 15419 . . . . . . . . . . . . 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)
88 simplrl 775 . . . . . . . . . . . . 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)
89 simprl 769 . . . . . . . . . . . . . . . . 17 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → 𝑓:𝑠⟶ℝ+)
9089ad2antrr 724 . . . . . . . . . . . . . . . 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+)
91 simprl 769 . . . . . . . . . . . . . . . 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)
9290, 91ffvelcdmd 7094 . . . . . . . . . . . . . . 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+)
9392rpred 13051 . . . . . . . . . . . . . 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)
9486, 43syl 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))
95 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧𝑢 = 𝑧)
96 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
9796oveq2d 7435 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓𝑢)) = (0(ball‘(abs ∘ − ))(𝑓𝑧)))
9897ineq1d 4209 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋))
9995, 98eqeq12d 2741 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋)))
100 simprr 771 . . . . . . . . . . . . . . . . . . . . 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))
101100ad2antrr 724 . . . . . . . . . . . . . . . . . . . 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))
10299, 101, 91rspcdva 3607 . . . . . . . . . . . . . . . . . . 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))
10385, 102eleqtrd 2827 . . . . . . . . . . . . . . . . . 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))
104103elin1d 4196 . . . . . . . . . . . . . . . . 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)))
105 0cnd 11239 . . . . . . . . . . . . . . . . . 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)
10692rpxrd 13052 . . . . . . . . . . . . . . . . . 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*)
107 elbl 24338 . . . . . . . . . . . . . . . . . 18 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (𝑓𝑧) ∈ ℝ*) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓𝑧)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓𝑧))))
10822, 105, 106, 107mp3an2i 1462 . . . . . . . . . . . . . . . . 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))))
109104, 108mpbid 231 . . . . . . . . . . . . . . . 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)))
110109simprd 494 . . . . . . . . . . . . . . 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))
11194, 110eqbrtrrd 5173 . . . . . . . . . . . . . 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))
11296breq1d 5159 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → ((𝑓𝑢) ≤ 𝑟 ↔ (𝑓𝑧) ≤ 𝑟))
113 simplrr 776 . . . . . . . . . . . . . . 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)
114112, 113, 91rspcdva 3607 . . . . . . . . . . . . . 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)
11587, 93, 88, 111, 114ltletrd 11406 . . . . . . . . . . . . 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)
11687, 88, 115ltled 11394 . . . . . . . . . . . 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)
117116rexlimdvaa 3145 . . . . . . . . . . 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))
11878, 117sylbid 239 . . . . . . . . . 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))
119118ralrimiv 3134 . . . . . . . . 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)
120 simpllr 774 . . . . . . . . . . 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))
121120elin2d 4197 . . . . . . . . . 10 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → 𝑠 ∈ Fin)
122 ffvelcdm 7090 . . . . . . . . . . . . 13 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ+)
123122rpred 13051 . . . . . . . . . . . 12 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ)
124123ralrimiva 3135 . . . . . . . . . . 11 (𝑓:𝑠⟶ℝ+ → ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ)
125124ad2antrl 726 . . . . . . . . . 10 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ)
126 fimaxre3 12193 . . . . . . . . . 10 ((𝑠 ∈ Fin ∧ ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)
127121, 125, 126syl2anc 582 . . . . . . . . 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)
128119, 127reximddv 3160 . . . . . . . 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)
129128ex 411 . . . . . . 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))
130129exlimdv 1928 . . . . . 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))
131130expimpd 452 . . . . 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))
132131rexlimdva 3144 . . . 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))
13372, 132mpd 15 . . 3 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)
13410, 133jca 510 . 2 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
135 eqid 2725 . . . . . 6 (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
136 eqid 2725 . . . . . 6 ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) “ ((-𝑟[,]𝑟) × (-𝑟[,]𝑟))) = ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) “ ((-𝑟[,]𝑟) × (-𝑟[,]𝑟)))
1371, 4, 135, 136cnheiborlem 24924 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
138137rexlimdvaa 3145 . . . 4 (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟𝑇 ∈ Comp))
139138imp 405 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
140139adantl 480 . 2 ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
141134, 140impbida 799 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 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  cin 3943  wss 3944  𝒫 cpw 4604   cuni 4909   class class class wbr 5149   × cxp 5676  cima 5681  ccom 5682  wf 6545  cfv 6549  (class class class)co 7419  cmpo 7421  Fincfn 8964  cc 11138  cr 11139  0cc0 11140  1c1 11141  ici 11142   + caddc 11143   · cmul 11145  *cxr 11279   < clt 11280  cle 11281  cmin 11476  -cneg 11477  +crp 13009  [,]cicc 13362  abscabs 15217  t crest 17405  TopOpenctopn 17406  ∞Metcxmet 21281  ballcbl 21283  fldccnfld 21296  Topctop 22839  Clsdccld 22964  Hauscha 23256  Compccmp 23334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218  ax-addf 11219
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-fi 9436  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-ioo 13363  df-icc 13366  df-fz 13520  df-fzo 13663  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-starv 17251  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-unif 17259  df-hom 17260  df-cco 17261  df-rest 17407  df-topn 17408  df-0g 17426  df-gsum 17427  df-topgen 17428  df-pt 17429  df-prds 17432  df-xrs 17487  df-qtop 17492  df-imas 17493  df-xps 17495  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-mulg 19032  df-cntz 19280  df-cmn 19749  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-cnfld 21297  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22893  df-cld 22967  df-cls 22969  df-cn 23175  df-cnp 23176  df-haus 23263  df-cmp 23335  df-tx 23510  df-hmeo 23703  df-xms 24270  df-ms 24271  df-tms 24272  df-cncf 24842
This theorem is referenced by:  cnllycmp  24926  cncmet  25294  ftalem3  27052
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