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Theorem cnheibor 22562
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 𝑓 𝑠 𝑢 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnheibor.2 . . . . . 6 𝐽 = (TopOpen‘ℂfld)
21cnfldhaus 22398 . . . . 5 𝐽 ∈ Haus
32a1i 11 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝐽 ∈ Haus)
4 simpl 472 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
5 cnheibor.3 . . . . 5 𝑇 = (𝐽t 𝑋)
6 simpr 476 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
75, 6syl5eqelr 2693 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽t 𝑋) ∈ Comp)
81cnfldtopon 22396 . . . . . 6 𝐽 ∈ (TopOn‘ℂ)
98toponunii 20547 . . . . 5 ℂ = 𝐽
109hauscmp 21020 . . . 4 ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
113, 4, 7, 10syl3anc 1318 . . 3 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
121cnfldtop 22397 . . . . . . . . . . 11 𝐽 ∈ Top
139restuni 20776 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = (𝐽t 𝑋))
1412, 4, 13sylancr 694 . . . . . . . . . 10 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = (𝐽t 𝑋))
155unieqi 4381 . . . . . . . . . 10 𝑇 = (𝐽t 𝑋)
1614, 15syl6eqr 2662 . . . . . . . . 9 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = 𝑇)
1716eleq2d 2673 . . . . . . . 8 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥𝑋𝑥 𝑇))
1817biimpar 501 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → 𝑥𝑋)
1912a1i 11 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝐽 ∈ Top)
20 cnex 9896 . . . . . . . . . . . 12 ℂ ∈ V
21 ssexg 4732 . . . . . . . . . . . 12 ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
224, 20, 21sylancl 693 . . . . . . . . . . 11 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
2322adantr 480 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑋 ∈ V)
24 cnxmet 22386 . . . . . . . . . . . 12 (abs ∘ − ) ∈ (∞Met‘ℂ)
2524a1i 11 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs ∘ − ) ∈ (∞Met‘ℂ))
26 0cnd 9912 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ∈ ℂ)
274sselda 3568 . . . . . . . . . . . . . 14 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ℂ)
2827abscld 14023 . . . . . . . . . . . . 13 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) ∈ ℝ)
29 peano2re 10088 . . . . . . . . . . . . 13 ((abs‘𝑥) ∈ ℝ → ((abs‘𝑥) + 1) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . . 12 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ)
3130rexrd 9968 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ*)
321cnfldtopn 22395 . . . . . . . . . . . 12 𝐽 = (MetOpen‘(abs ∘ − ))
3332blopn 22115 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
3425, 26, 31, 33syl3anc 1318 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
35 elrestr 15912 . . . . . . . . . 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 1318 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽t 𝑋))
3736, 5syl6eleqr 2699 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
38 0cn 9911 . . . . . . . . . . . . . 14 0 ∈ ℂ
39 eqid 2610 . . . . . . . . . . . . . . 15 (abs ∘ − ) = (abs ∘ − )
4039cnmetdval 22384 . . . . . . . . . . . . . 14 ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
4138, 40mpan 702 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
42 df-neg 10148 . . . . . . . . . . . . . . 15 -𝑥 = (0 − 𝑥)
4342fveq2i 6106 . . . . . . . . . . . . . 14 (abs‘-𝑥) = (abs‘(0 − 𝑥))
44 absneg 13865 . . . . . . . . . . . . . 14 (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
4543, 44syl5eqr 2658 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
4641, 45eqtrd 2644 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4727, 46syl 17 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
4828ltp1d 10833 . . . . . . . . . . 11 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
4947, 48eqbrtrd 4605 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
50 elbl 22003 . . . . . . . . . . 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 1318 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → (𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))))
5227, 49, 51mpbir2and 959 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
53 simpr 476 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥𝑋)
5452, 53elind 3760 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
5527absge0d 14031 . . . . . . . . . 10 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → 0 ≤ (abs‘𝑥))
5628, 55ge0p1rpd 11778 . . . . . . . . 9 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ((abs‘𝑥) + 1) ∈ ℝ+)
57 eqid 2610 . . . . . . . . 9 ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
58 oveq2 6557 . . . . . . . . . . . 12 (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
5958ineq1d 3775 . . . . . . . . . . 11 (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
6059eqeq2d 2620 . . . . . . . . . 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 3282 . . . . . . . . 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 693 . . . . . . . 8 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ∃𝑟 ∈ ℝ+ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
63 eleq2 2677 . . . . . . . . . 10 (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥𝑢𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
64 eqeq1 2614 . . . . . . . . . . 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 3034 . . . . . . . . . 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 743 . . . . . . . . 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 3282 . . . . . . . 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 1316 . . . . . . 7 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥𝑋) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
6918, 68syldan 486 . . . . . 6 (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 𝑇) → ∃𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
7069ralrimiva 2949 . . . . 5 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 𝑇𝑢𝑇 (𝑥𝑢 ∧ ∃𝑟 ∈ ℝ+ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
71 eqid 2610 . . . . . 6 𝑇 = 𝑇
72 oveq2 6557 . . . . . . . 8 (𝑟 = (𝑓𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓𝑢)))
7372ineq1d 3775 . . . . . . 7 (𝑟 = (𝑓𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))
7473eqeq2d 2620 . . . . . 6 (𝑟 = (𝑓𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋)))
7571, 74cmpcovf 21004 . . . . 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 691 . . . 4 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)( 𝑇 = 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))))
7716ad4antr 764 . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . 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 2673 . . . . . . . . . . . 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 4376 . . . . . . . . . . . 12 (𝑥 𝑠 ↔ ∃𝑧𝑠 𝑥𝑧)
8280, 81syl6bb 275 . . . . . . . . . . 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 4403 . . . . . . . . . . . . . . . . . 18 (𝑧𝑠𝑧 𝑠)
8483ad2antrl 760 . . . . . . . . . . . . . . . . 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 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)
8684, 85sseqtr4d 3605 . . . . . . . . . . . . . . . 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 806 . . . . . . . . . . . . . . . 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 3578 . . . . . . . . . . . . . . 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 792 . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . 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 14023 . . . . . . . . . . . . 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 796 . . . . . . . . . . . . 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 790 . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . 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 790 . . . . . . . . . . . . . . . 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 6268 . . . . . . . . . . . . . . 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 11748 . . . . . . . . . . . . . 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 3795 . . . . . . . . . . . . . . . . . 18 ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋) ⊆ (0(ball‘(abs ∘ − ))(𝑓𝑧))
100 simprr 792 . . . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . 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))
102 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑧𝑢 = 𝑧)
103 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
104103oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓𝑢)) = (0(ball‘(abs ∘ − ))(𝑓𝑧)))
105104ineq1d 3775 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋))
106102, 105eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋)))
107106rspcv 3278 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑠 → (∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋) → 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓𝑧)) ∩ 𝑋)))
10895, 101, 107sylc 63 . . . . . . . . . . . . . . . . . . 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))
10989, 108eleqtrd 2690 . . . . . . . . . . . . . . . . . 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))
11099, 109sseldi 3566 . . . . . . . . . . . . . . . . 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)))
11124a1i 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))
112 0cnd 9912 . . . . . . . . . . . . . . . . . 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)
11396rpxrd 11749 . . . . . . . . . . . . . . . . . 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*)
114 elbl 22003 . . . . . . . . . . . . . . . . . 18 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (𝑓𝑧) ∈ ℝ*) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓𝑧)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓𝑧))))
115111, 112, 113, 114syl3anc 1318 . . . . . . . . . . . . . . . . 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))))
116110, 115mpbid 221 . . . . . . . . . . . . . . . 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)))
117116simprd 478 . . . . . . . . . . . . . . 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))
11898, 117eqbrtrrd 4607 . . . . . . . . . . . . . 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))
119 simplrr 797 . . . . . . . . . . . . . . 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)
120103breq1d 4593 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑧 → ((𝑓𝑢) ≤ 𝑟 ↔ (𝑓𝑧) ≤ 𝑟))
121120rspcv 3278 . . . . . . . . . . . . . . 15 (𝑧𝑠 → (∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟 → (𝑓𝑧) ≤ 𝑟))
12295, 119, 121sylc 63 . . . . . . . . . . . . . 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)
12391, 97, 92, 118, 122ltletrd 10076 . . . . . . . . . . . . 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)
12491, 92, 123ltled 10064 . . . . . . . . . . . 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)
125124rexlimdvaa 3014 . . . . . . . . . . 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))
12682, 125sylbid 229 . . . . . . . . . 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))
127126ralrimiv 2948 . . . . . . . . 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)
128 inss2 3796 . . . . . . . . . . 11 (𝒫 𝑇 ∩ Fin) ⊆ Fin
129 simpllr 795 . . . . . . . . . . 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))
130128, 129sseldi 3566 . . . . . . . . . 10 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → 𝑠 ∈ Fin)
131 ffvelrn 6265 . . . . . . . . . . . . 13 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ+)
132131rpred 11748 . . . . . . . . . . . 12 ((𝑓:𝑠⟶ℝ+𝑢𝑠) → (𝑓𝑢) ∈ ℝ)
133132ralrimiva 2949 . . . . . . . . . . 11 (𝑓:𝑠⟶ℝ+ → ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ)
134133ad2antrl 760 . . . . . . . . . 10 (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ 𝑇 = 𝑠) ∧ (𝑓:𝑠⟶ℝ+ ∧ ∀𝑢𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓𝑢)) ∩ 𝑋))) → ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ)
135 fimaxre3 10849 . . . . . . . . . 10 ((𝑠 ∈ Fin ∧ ∀𝑢𝑠 (𝑓𝑢) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢𝑠 (𝑓𝑢) ≤ 𝑟)
136130, 134, 135syl2anc 691 . . . . . . . . 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)
137127, 136reximddv 3001 . . . . . . . 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)
138137ex 449 . . . . . . 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))
139138exlimdv 1848 . . . . . 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))
140139expimpd 627 . . . . 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))
141140rexlimdva 3013 . . . 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))
14276, 141mpd 15 . . 3 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)
14311, 142jca 553 . 2 ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟))
144 eqid 2610 . . . . . 6 (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
145 eqid 2610 . . . . . 6 ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) “ ((-𝑟[,]𝑟) × (-𝑟[,]𝑟))) = ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) “ ((-𝑟[,]𝑟) × (-𝑟[,]𝑟)))
1461, 5, 144, 145cnheiborlem 22561 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
147146rexlimdvaa 3014 . . . 4 (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟𝑇 ∈ Comp))
148147imp 444 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
149148adantl 481 . 2 ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
150143, 149impbida 873 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 195  wa 383   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4372   class class class wbr 4583   × cxp 5036  cima 5041  ccom 5042  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  Fincfn 7841  cc 9813  cr 9814  0cc0 9815  1c1 9816  ici 9817   + caddc 9818   · cmul 9820  *cxr 9952   < clt 9953  cle 9954  cmin 10145  -cneg 10146  +crp 11708  [,]cicc 12049  abscabs 13822  t crest 15904  TopOpenctopn 15905  ∞Metcxmt 19552  ballcbl 19554  fldccnfld 19567  Topctop 20517  Clsdccld 20630  Hauscha 20922  Compccmp 20999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-cls 20635  df-cn 20841  df-cnp 20842  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489
This theorem is referenced by:  cnllycmp  22563  cncmet  22927  ftalem3  24601
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