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Theorem cnheibor 24987
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 24805 . . . 4 𝐽 ∈ Haus
3 simpl 482 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
4 cnheibor.3 . . . . 5 𝑇 = (𝐽t 𝑋)
5 simpr 484 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
64, 5eqeltrrid 2846 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽t 𝑋) ∈ Comp)
71cnfldtopon 24803 . . . . . 6 𝐽 ∈ (TopOn‘ℂ)
87toponunii 22922 . . . . 5 ℂ = 𝐽
98hauscmp 23415 . . . 4 ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
102, 3, 6, 9mp3an2i 1468 . . 3 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
111cnfldtop 24804 . . . . . . . . . . 11 𝐽 ∈ Top
128restuni 23170 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = (𝐽t 𝑋))
1311, 3, 12sylancr 587 . . . . . . . . . 10 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = (𝐽t 𝑋))
144unieqi 4919 . . . . . . . . . 10 𝑇 = (𝐽t 𝑋)
1513, 14eqtr4di 2795 . . . . . . . . 9 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = 𝑇)
1615eleq2d 2827 . . . . . . . 8 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥𝑋𝑥 𝑇))
1716biimpar 477 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → 𝑥𝑋)
18 cnex 11236 . . . . . . . . . . . 12 ℂ ∈ V
19 ssexg 5323 . . . . . . . . . . . 12 ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
203, 18, 19sylancl 586 . . . . . . . . . . 11 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
2120adantr 480 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑋 ∈ V)
22 cnxmet 24793 . . . . . . . . . . 11 (abs ∘ − ) ∈ (∞Met‘ℂ)
23 0cnd 11254 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ∈ ℂ)
243sselda 3983 . . . . . . . . . . . . . 14 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ℂ)
2524abscld 15475 . . . . . . . . . . . . 13 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) ∈ ℝ)
26 peano2re 11434 . . . . . . . . . . . . 13 ((abs‘𝑥) ∈ ℝ → ((abs‘𝑥) + 1) ∈ ℝ)
2725, 26syl 17 . . . . . . . . . . . 12 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ)
2827rexrd 11311 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ*)
291cnfldtopn 24802 . . . . . . . . . . . 12 𝐽 = (MetOpen‘(abs ∘ − ))
3029blopn 24513 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
3122, 23, 28, 30mp3an2i 1468 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
32 elrestr 17473 . . . . . . . . . 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 1468 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽t 𝑋))
3433, 4eleqtrrdi 2852 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
35 0cn 11253 . . . . . . . . . . . . . 14 0 ∈ ℂ
36 eqid 2737 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
3736cnmetdval 24791 . . . . . . . . . . . . . 14 ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
3835, 37mpan 690 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
39 df-neg 11495 . . . . . . . . . . . . . . 15 -𝑥 = (0 − 𝑥)
4039fveq2i 6909 . . . . . . . . . . . . . 14 (abs‘-𝑥) = (abs‘(0 − 𝑥))
41 absneg 15316 . . . . . . . . . . . . . 14 (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
4240, 41eqtr3id 2791 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
4338, 42eqtrd 2777 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4424, 43syl 17 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4525ltp1d 12198 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
4644, 45eqbrtrd 5165 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
47 elbl 24398 . . . . . . . . . . 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 1468 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))))
4924, 46, 48mpbir2and 713 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
50 simpr 484 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥𝑋)
5149, 50elind 4200 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
5224absge0d 15483 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ≤ (abs‘𝑥))
5325, 52ge0p1rpd 13107 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ+)
54 eqid 2737 . . . . . . . . 9 ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
55 oveq2 7439 . . . . . . . . . . 11 (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
5655ineq1d 4219 . . . . . . . . . 10 (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
5756rspceeqv 3645 . . . . . . . . 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 586 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ∃𝑟 ∈ ℝ+ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
59 eleq2 2830 . . . . . . . . . 10 (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥𝑢𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
60 eqeq1 2741 . . . . . . . . . . 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 3179 . . . . . . . . . 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 632 . . . . . . . . 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 3622 . . . . . . . 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 837 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6517, 64syldan 591 . . . . . 6 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6665ralrimiva 3146 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 𝑇𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
67 eqid 2737 . . . . . 6 𝑇 = 𝑇
68 oveq2 7439 . . . . . . . 8 (𝑟 = (𝑓𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓𝑢)))
6968ineq1d 4219 . . . . . . 7 (𝑟 = (𝑓𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))
7069eqeq2d 2748 . . . . . 6 (𝑟 = (𝑓𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋)))
7167, 70cmpcovf 23399 . . . . 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 584 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)( 𝑇 = 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))))
7315ad4antr 732 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 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 2827 . . . . . . . . . . . 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 4911 . . . . . . . . . . . 12 (𝑥 𝑠 ↔ ∃𝑧𝑠 𝑥𝑧)
7876, 77bitrdi 287 . . . . . . . . . . 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 4937 . . . . . . . . . . . . . . . . . 18 (𝑧𝑠𝑧 𝑠)
8079ad2antrl 728 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 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 4021 . . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . 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 3994 . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . 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 15475 . . . . . . . . . . . . 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 777 . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . 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 13077 . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
9796oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓𝑢)) = (0(ball‘(abs ∘ − ))(𝑓𝑧)))
9897ineq1d 4219 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋))
9995, 98eqeq12d 2753 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋)))
100 simprr 773 . . . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . 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 3623 . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . 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 4204 . . . . . . . . . . . . . . . . 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 11254 . . . . . . . . . . . . . . . . . 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 13078 . . . . . . . . . . . . . . . . . 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 24398 . . . . . . . . . . . . . . . . . 18 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (𝑓𝑧) ∈ ℝ*) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓𝑧)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓𝑧))))
10822, 105, 106, 107mp3an2i 1468 . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 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 5167 . . . . . . . . . . . . . 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 5153 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → ((𝑓𝑢) ≤ 𝑟 ↔ (𝑓𝑧) ≤ 𝑟))
113 simplrr 778 . . . . . . . . . . . . . . 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 3623 . . . . . . . . . . . . . 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 11421 . . . . . . . . . . . . 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 11409 . . . . . . . . . . . 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 3156 . . . . . . . . . . 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 240 . . . . . . . . . 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 3145 . . . . . . . . 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 776 . . . . . . . . . . 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 4205 . . . . . . . . . 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 7101 . . . . . . . . . . . . 13 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ+)
123122rpred 13077 . . . . . . . . . . . 12 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ)
124123ralrimiva 3146 . . . . . . . . . . 11 (𝑓:𝑠⟶ℝ+ → ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ)
125124ad2antrl 728 . . . . . . . . . 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 12214 . . . . . . . . . 10 ((𝑠 ∈ Fin ∧ ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)
127121, 125, 126syl2anc 584 . . . . . . . . 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 3171 . . . . . . . 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 412 . . . . . . 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 1933 . . . . . 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 453 . . . . 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 3155 . . . 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 511 . 2 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
135 eqid 2737 . . . . . 6 (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
136 eqid 2737 . . . . . 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 24986 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
138137rexlimdvaa 3156 . . . 4 (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟𝑇 ∈ Comp))
139138imp 406 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
140139adantl 481 . 2 ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
141134, 140impbida 801 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 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143   × cxp 5683  cima 5688  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156  ici 11157   + caddc 11158   · cmul 11160  *cxr 11294   < clt 11295  cle 11296  cmin 11492  -cneg 11493  +crp 13034  [,]cicc 13390  abscabs 15273  t crest 17465  TopOpenctopn 17466  ∞Metcxmet 21349  ballcbl 21351  fldccnfld 21364  Topctop 22899  Clsdccld 23024  Hauscha 23316  Compccmp 23394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-cls 23029  df-cn 23235  df-cnp 23236  df-haus 23323  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904
This theorem is referenced by:  cnllycmp  24988  cncmet  25356  ftalem3  27118
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