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Theorem cnheibor 24118
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 23948 . . . 4 𝐽 ∈ Haus
3 simpl 483 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
4 cnheibor.3 . . . . 5 𝑇 = (𝐽t 𝑋)
5 simpr 485 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
64, 5eqeltrrid 2844 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽t 𝑋) ∈ Comp)
71cnfldtopon 23946 . . . . . 6 𝐽 ∈ (TopOn‘ℂ)
87toponunii 22065 . . . . 5 ℂ = 𝐽
98hauscmp 22558 . . . 4 ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
102, 3, 6, 9mp3an2i 1465 . . 3 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
111cnfldtop 23947 . . . . . . . . . . 11 𝐽 ∈ Top
128restuni 22313 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = (𝐽t 𝑋))
1311, 3, 12sylancr 587 . . . . . . . . . 10 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = (𝐽t 𝑋))
144unieqi 4852 . . . . . . . . . 10 𝑇 = (𝐽t 𝑋)
1513, 14eqtr4di 2796 . . . . . . . . 9 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = 𝑇)
1615eleq2d 2824 . . . . . . . 8 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥𝑋𝑥 𝑇))
1716biimpar 478 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → 𝑥𝑋)
18 cnex 10952 . . . . . . . . . . . 12 ℂ ∈ V
19 ssexg 5247 . . . . . . . . . . . 12 ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
203, 18, 19sylancl 586 . . . . . . . . . . 11 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
2120adantr 481 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑋 ∈ V)
22 cnxmet 23936 . . . . . . . . . . 11 (abs ∘ − ) ∈ (∞Met‘ℂ)
23 0cnd 10968 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ∈ ℂ)
243sselda 3921 . . . . . . . . . . . . . 14 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ℂ)
2524abscld 15148 . . . . . . . . . . . . 13 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) ∈ ℝ)
26 peano2re 11148 . . . . . . . . . . . . 13 ((abs‘𝑥) ∈ ℝ → ((abs‘𝑥) + 1) ∈ ℝ)
2725, 26syl 17 . . . . . . . . . . . 12 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ)
2827rexrd 11025 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ*)
291cnfldtopn 23945 . . . . . . . . . . . 12 𝐽 = (MetOpen‘(abs ∘ − ))
3029blopn 23656 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
3122, 23, 28, 30mp3an2i 1465 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
32 elrestr 17139 . . . . . . . . . 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 1465 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽t 𝑋))
3433, 4eleqtrrdi 2850 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
35 0cn 10967 . . . . . . . . . . . . . 14 0 ∈ ℂ
36 eqid 2738 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
3736cnmetdval 23934 . . . . . . . . . . . . . 14 ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
3835, 37mpan 687 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
39 df-neg 11208 . . . . . . . . . . . . . . 15 -𝑥 = (0 − 𝑥)
4039fveq2i 6777 . . . . . . . . . . . . . 14 (abs‘-𝑥) = (abs‘(0 − 𝑥))
41 absneg 14989 . . . . . . . . . . . . . 14 (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
4240, 41eqtr3id 2792 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
4338, 42eqtrd 2778 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4424, 43syl 17 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4525ltp1d 11905 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
4644, 45eqbrtrd 5096 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
47 elbl 23541 . . . . . . . . . . 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 1465 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))))
4924, 46, 48mpbir2and 710 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
50 simpr 485 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥𝑋)
5149, 50elind 4128 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
5224absge0d 15156 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ≤ (abs‘𝑥))
5325, 52ge0p1rpd 12802 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ+)
54 eqid 2738 . . . . . . . . 9 ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
55 oveq2 7283 . . . . . . . . . . 11 (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
5655ineq1d 4145 . . . . . . . . . 10 (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
5756rspceeqv 3575 . . . . . . . . 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 2827 . . . . . . . . . 10 (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥𝑢𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
60 eqeq1 2742 . . . . . . . . . . 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 3226 . . . . . . . . . 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 631 . . . . . . . . 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 3561 . . . . . . . 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 834 . . . . . . 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 3103 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 𝑇𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
67 eqid 2738 . . . . . 6 𝑇 = 𝑇
68 oveq2 7283 . . . . . . . 8 (𝑟 = (𝑓𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓𝑢)))
6968ineq1d 4145 . . . . . . 7 (𝑟 = (𝑓𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))
7069eqeq2d 2749 . . . . . 6 (𝑟 = (𝑓𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋)))
7167, 70cmpcovf 22542 . . . . 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 729 . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . 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 2824 . . . . . . . . . . . 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 4843 . . . . . . . . . . . 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 4871 . . . . . . . . . . . . . . . . . 18 (𝑧𝑠𝑧 𝑠)
8079ad2antrl 725 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 3962 . . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . . 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 3931 . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . 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 3922 . . . . . . . . . . . . . 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 15148 . . . . . . . . . . . . 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 774 . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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, 91ffvelrnd 6962 . . . . . . . . . . . . . . 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 12772 . . . . . . . . . . . . . 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 6774 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
9796oveq2d 7291 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓𝑢)) = (0(ball‘(abs ∘ − ))(𝑓𝑧)))
9897ineq1d 4145 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋))
9995, 98eqeq12d 2754 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋)))
100 simprr 770 . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . 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 3562 . . . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . . 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 10968 . . . . . . . . . . . . . . . . . 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 12773 . . . . . . . . . . . . . . . . . 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 23541 . . . . . . . . . . . . . . . . . 18 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (𝑓𝑧) ∈ ℝ*) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓𝑧)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓𝑧))))
10822, 105, 106, 107mp3an2i 1465 . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . 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 5098 . . . . . . . . . . . . . 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 5084 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → ((𝑓𝑢) ≤ 𝑟 ↔ (𝑓𝑧) ≤ 𝑟))
113 simplrr 775 . . . . . . . . . . . . . . 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 3562 . . . . . . . . . . . . . 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 11135 . . . . . . . . . . . . 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 11123 . . . . . . . . . . . 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 3214 . . . . . . . . . . 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 3102 . . . . . . . . 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 773 . . . . . . . . . . 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 4133 . . . . . . . . . 10 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → 𝑠 ∈ Fin)
122 ffvelrn 6959 . . . . . . . . . . . . 13 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ+)
123122rpred 12772 . . . . . . . . . . . 12 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ)
124123ralrimiva 3103 . . . . . . . . . . 11 (𝑓:𝑠⟶ℝ+ → ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ)
125124ad2antrl 725 . . . . . . . . . 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 11921 . . . . . . . . . 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 3204 . . . . . . . 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 413 . . . . . . 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 1936 . . . . . 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 454 . . . . 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 3213 . . . 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 512 . 2 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
135 eqid 2738 . . . . . 6 (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
136 eqid 2738 . . . . . 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 24117 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
138137rexlimdvaa 3214 . . . 4 (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟𝑇 ∈ Comp))
139138imp 407 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
140139adantl 482 . 2 ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
141134, 140impbida 798 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 396   = wceq 1539  wex 1782  wcel 2106  wral 3064  wrex 3065  Vcvv 3432  cin 3886  wss 3887  𝒫 cpw 4533   cuni 4839   class class class wbr 5074   × cxp 5587  cima 5592  ccom 5593  wf 6429  cfv 6433  (class class class)co 7275  cmpo 7277  Fincfn 8733  cc 10869  cr 10870  0cc0 10871  1c1 10872  ici 10873   + caddc 10874   · cmul 10876  *cxr 11008   < clt 11009  cle 11010  cmin 11205  -cneg 11206  +crp 12730  [,]cicc 13082  abscabs 14945  t crest 17131  TopOpenctopn 17132  ∞Metcxmet 20582  ballcbl 20584  fldccnfld 20597  Topctop 22042  Clsdccld 22167  Hauscha 22459  Compccmp 22537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-cls 22172  df-cn 22378  df-cnp 22379  df-haus 22466  df-cmp 22538  df-tx 22713  df-hmeo 22906  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041
This theorem is referenced by:  cnllycmp  24119  cncmet  24486  ftalem3  26224
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