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Theorem cnheibor 24836
Description: Heine-Borel theorem for complex numbers. A subset of β„‚ is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
cnheibor.2 𝐽 = (TopOpenβ€˜β„‚fld)
cnheibor.3 𝑇 = (𝐽 β†Ύt 𝑋)
Assertion
Ref Expression
cnheibor (𝑋 βŠ† β„‚ β†’ (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)))
Distinct variable groups:   π‘₯,π‘Ÿ,𝑇   𝐽,π‘Ÿ,π‘₯   𝑋,π‘Ÿ,π‘₯

Proof of Theorem cnheibor
Dummy variables 𝑧 𝑒 𝑓 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnheibor.2 . . . . 5 𝐽 = (TopOpenβ€˜β„‚fld)
21cnfldhaus 24656 . . . 4 𝐽 ∈ Haus
3 simpl 482 . . . 4 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 βŠ† β„‚)
4 cnheibor.3 . . . . 5 𝑇 = (𝐽 β†Ύt 𝑋)
5 simpr 484 . . . . 5 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑇 ∈ Comp)
64, 5eqeltrrid 2832 . . . 4 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ (𝐽 β†Ύt 𝑋) ∈ Comp)
71cnfldtopon 24654 . . . . . 6 𝐽 ∈ (TopOnβ€˜β„‚)
87toponunii 22773 . . . . 5 β„‚ = βˆͺ 𝐽
98hauscmp 23266 . . . 4 ((𝐽 ∈ Haus ∧ 𝑋 βŠ† β„‚ ∧ (𝐽 β†Ύt 𝑋) ∈ Comp) β†’ 𝑋 ∈ (Clsdβ€˜π½))
102, 3, 6, 9mp3an2i 1462 . . 3 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 ∈ (Clsdβ€˜π½))
111cnfldtop 24655 . . . . . . . . . . 11 𝐽 ∈ Top
128restuni 23021 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 βŠ† β„‚) β†’ 𝑋 = βˆͺ (𝐽 β†Ύt 𝑋))
1311, 3, 12sylancr 586 . . . . . . . . . 10 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 = βˆͺ (𝐽 β†Ύt 𝑋))
144unieqi 4914 . . . . . . . . . 10 βˆͺ 𝑇 = βˆͺ (𝐽 β†Ύt 𝑋)
1513, 14eqtr4di 2784 . . . . . . . . 9 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 = βˆͺ 𝑇)
1615eleq2d 2813 . . . . . . . 8 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝑇))
1716biimpar 477 . . . . . . 7 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ βˆͺ 𝑇) β†’ π‘₯ ∈ 𝑋)
18 cnex 11193 . . . . . . . . . . . 12 β„‚ ∈ V
19 ssexg 5316 . . . . . . . . . . . 12 ((𝑋 βŠ† β„‚ ∧ β„‚ ∈ V) β†’ 𝑋 ∈ V)
203, 18, 19sylancl 585 . . . . . . . . . . 11 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 ∈ V)
2120adantr 480 . . . . . . . . . 10 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 ∈ V)
22 cnxmet 24644 . . . . . . . . . . 11 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
23 0cnd 11211 . . . . . . . . . . 11 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ 0 ∈ β„‚)
243sselda 3977 . . . . . . . . . . . . . 14 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ β„‚)
2524abscld 15389 . . . . . . . . . . . . 13 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π‘₯) ∈ ℝ)
26 peano2re 11391 . . . . . . . . . . . . 13 ((absβ€˜π‘₯) ∈ ℝ β†’ ((absβ€˜π‘₯) + 1) ∈ ℝ)
2725, 26syl 17 . . . . . . . . . . . 12 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ ((absβ€˜π‘₯) + 1) ∈ ℝ)
2827rexrd 11268 . . . . . . . . . . 11 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ ((absβ€˜π‘₯) + 1) ∈ ℝ*)
291cnfldtopn 24653 . . . . . . . . . . . 12 𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))
3029blopn 24364 . . . . . . . . . . 11 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 0 ∈ β„‚ ∧ ((absβ€˜π‘₯) + 1) ∈ ℝ*) β†’ (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∈ 𝐽)
3122, 23, 28, 30mp3an2i 1462 . . . . . . . . . 10 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∈ 𝐽)
32 elrestr 17383 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 ∈ V ∧ (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∈ 𝐽) β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) ∈ (𝐽 β†Ύt 𝑋))
3311, 21, 31, 32mp3an2i 1462 . . . . . . . . 9 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) ∈ (𝐽 β†Ύt 𝑋))
3433, 4eleqtrrdi 2838 . . . . . . . 8 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) ∈ 𝑇)
35 0cn 11210 . . . . . . . . . . . . . 14 0 ∈ β„‚
36 eqid 2726 . . . . . . . . . . . . . . 15 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
3736cnmetdval 24642 . . . . . . . . . . . . . 14 ((0 ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ (0(abs ∘ βˆ’ )π‘₯) = (absβ€˜(0 βˆ’ π‘₯)))
3835, 37mpan 687 . . . . . . . . . . . . 13 (π‘₯ ∈ β„‚ β†’ (0(abs ∘ βˆ’ )π‘₯) = (absβ€˜(0 βˆ’ π‘₯)))
39 df-neg 11451 . . . . . . . . . . . . . . 15 -π‘₯ = (0 βˆ’ π‘₯)
4039fveq2i 6888 . . . . . . . . . . . . . 14 (absβ€˜-π‘₯) = (absβ€˜(0 βˆ’ π‘₯))
41 absneg 15230 . . . . . . . . . . . . . 14 (π‘₯ ∈ β„‚ β†’ (absβ€˜-π‘₯) = (absβ€˜π‘₯))
4240, 41eqtr3id 2780 . . . . . . . . . . . . 13 (π‘₯ ∈ β„‚ β†’ (absβ€˜(0 βˆ’ π‘₯)) = (absβ€˜π‘₯))
4338, 42eqtrd 2766 . . . . . . . . . . . 12 (π‘₯ ∈ β„‚ β†’ (0(abs ∘ βˆ’ )π‘₯) = (absβ€˜π‘₯))
4424, 43syl 17 . . . . . . . . . . 11 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (0(abs ∘ βˆ’ )π‘₯) = (absβ€˜π‘₯))
4525ltp1d 12148 . . . . . . . . . . 11 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π‘₯) < ((absβ€˜π‘₯) + 1))
4644, 45eqbrtrd 5163 . . . . . . . . . 10 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (0(abs ∘ βˆ’ )π‘₯) < ((absβ€˜π‘₯) + 1))
47 elbl 24249 . . . . . . . . . . 11 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 0 ∈ β„‚ ∧ ((absβ€˜π‘₯) + 1) ∈ ℝ*) β†’ (π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ↔ (π‘₯ ∈ β„‚ ∧ (0(abs ∘ βˆ’ )π‘₯) < ((absβ€˜π‘₯) + 1))))
4822, 23, 28, 47mp3an2i 1462 . . . . . . . . . 10 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ↔ (π‘₯ ∈ β„‚ ∧ (0(abs ∘ βˆ’ )π‘₯) < ((absβ€˜π‘₯) + 1))))
4924, 46, 48mpbir2and 710 . . . . . . . . 9 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)))
50 simpr 484 . . . . . . . . 9 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
5149, 50elind 4189 . . . . . . . 8 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋))
5224absge0d 15397 . . . . . . . . . 10 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ 0 ≀ (absβ€˜π‘₯))
5325, 52ge0p1rpd 13052 . . . . . . . . 9 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ ((absβ€˜π‘₯) + 1) ∈ ℝ+)
54 eqid 2726 . . . . . . . . 9 ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋)
55 oveq2 7413 . . . . . . . . . . 11 (π‘Ÿ = ((absβ€˜π‘₯) + 1) β†’ (0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) = (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)))
5655ineq1d 4206 . . . . . . . . . 10 (π‘Ÿ = ((absβ€˜π‘₯) + 1) β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋))
5756rspceeqv 3628 . . . . . . . . 9 ((((absβ€˜π‘₯) + 1) ∈ ℝ+ ∧ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ ℝ+ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋))
5853, 54, 57sylancl 585 . . . . . . . 8 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘Ÿ ∈ ℝ+ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋))
59 eleq2 2816 . . . . . . . . . 10 (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) β†’ (π‘₯ ∈ 𝑒 ↔ π‘₯ ∈ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋)))
60 eqeq1 2730 . . . . . . . . . . 11 (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) β†’ (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋) ↔ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋)))
6160rexbidv 3172 . . . . . . . . . 10 (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋) ↔ βˆƒπ‘Ÿ ∈ ℝ+ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋)))
6259, 61anbi12d 630 . . . . . . . . 9 (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) β†’ ((π‘₯ ∈ 𝑒 ∧ βˆƒπ‘Ÿ ∈ ℝ+ 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋)) ↔ (π‘₯ ∈ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) ∧ βˆƒπ‘Ÿ ∈ ℝ+ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋))))
6362rspcev 3606 . . . . . . . 8 ((((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) ∈ 𝑇 ∧ (π‘₯ ∈ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) ∧ βˆƒπ‘Ÿ ∈ ℝ+ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋))) β†’ βˆƒπ‘’ ∈ 𝑇 (π‘₯ ∈ 𝑒 ∧ βˆƒπ‘Ÿ ∈ ℝ+ 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋)))
6434, 51, 58, 63syl12anc 834 . . . . . . 7 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘’ ∈ 𝑇 (π‘₯ ∈ 𝑒 ∧ βˆƒπ‘Ÿ ∈ ℝ+ 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋)))
6517, 64syldan 590 . . . . . 6 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ βˆͺ 𝑇) β†’ βˆƒπ‘’ ∈ 𝑇 (π‘₯ ∈ 𝑒 ∧ βˆƒπ‘Ÿ ∈ ℝ+ 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋)))
6665ralrimiva 3140 . . . . 5 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ βˆ€π‘₯ ∈ βˆͺ π‘‡βˆƒπ‘’ ∈ 𝑇 (π‘₯ ∈ 𝑒 ∧ βˆƒπ‘Ÿ ∈ ℝ+ 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋)))
67 eqid 2726 . . . . . 6 βˆͺ 𝑇 = βˆͺ 𝑇
68 oveq2 7413 . . . . . . . 8 (π‘Ÿ = (π‘“β€˜π‘’) β†’ (0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) = (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)))
6968ineq1d 4206 . . . . . . 7 (π‘Ÿ = (π‘“β€˜π‘’) β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))
7069eqeq2d 2737 . . . . . 6 (π‘Ÿ = (π‘“β€˜π‘’) β†’ (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋) ↔ 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋)))
7167, 70cmpcovf 23250 . . . . 5 ((𝑇 ∈ Comp ∧ βˆ€π‘₯ ∈ βˆͺ π‘‡βˆƒπ‘’ ∈ 𝑇 (π‘₯ ∈ 𝑒 ∧ βˆƒπ‘Ÿ ∈ ℝ+ 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋))) β†’ βˆƒπ‘  ∈ (𝒫 𝑇 ∩ Fin)(βˆͺ 𝑇 = βˆͺ 𝑠 ∧ βˆƒπ‘“(𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))))
725, 66, 71syl2anc 583 . . . 4 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ βˆƒπ‘  ∈ (𝒫 𝑇 ∩ Fin)(βˆͺ 𝑇 = βˆͺ 𝑠 ∧ βˆƒπ‘“(𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))))
7315ad4antr 729 . . . . . . . . . . . . . 14 ((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) β†’ 𝑋 = βˆͺ 𝑇)
74 simpllr 773 . . . . . . . . . . . . . 14 ((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) β†’ βˆͺ 𝑇 = βˆͺ 𝑠)
7573, 74eqtrd 2766 . . . . . . . . . . . . 13 ((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) β†’ 𝑋 = βˆͺ 𝑠)
7675eleq2d 2813 . . . . . . . . . . . 12 ((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝑠))
77 eluni2 4906 . . . . . . . . . . . 12 (π‘₯ ∈ βˆͺ 𝑠 ↔ βˆƒπ‘§ ∈ 𝑠 π‘₯ ∈ 𝑧)
7876, 77bitrdi 287 . . . . . . . . . . 11 ((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) β†’ (π‘₯ ∈ 𝑋 ↔ βˆƒπ‘§ ∈ 𝑠 π‘₯ ∈ 𝑧))
79 elssuni 4934 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ 𝑠 β†’ 𝑧 βŠ† βˆͺ 𝑠)
8079ad2antrl 725 . . . . . . . . . . . . . . . . 17 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑧 βŠ† βˆͺ 𝑠)
8175adantr 480 . . . . . . . . . . . . . . . . 17 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑋 = βˆͺ 𝑠)
8280, 81sseqtrrd 4018 . . . . . . . . . . . . . . . 16 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑧 βŠ† 𝑋)
83 simp-6l 784 . . . . . . . . . . . . . . . 16 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑋 βŠ† β„‚)
8482, 83sstrd 3987 . . . . . . . . . . . . . . 15 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑧 βŠ† β„‚)
85 simprr 770 . . . . . . . . . . . . . . 15 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ π‘₯ ∈ 𝑧)
8684, 85sseldd 3978 . . . . . . . . . . . . . 14 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ π‘₯ ∈ β„‚)
8786abscld 15389 . . . . . . . . . . . . 13 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (absβ€˜π‘₯) ∈ ℝ)
88 simplrl 774 . . . . . . . . . . . . 13 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ π‘Ÿ ∈ ℝ)
89 simprl 768 . . . . . . . . . . . . . . . . 17 (((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ 𝑓:π‘ βŸΆβ„+)
9089ad2antrr 723 . . . . . . . . . . . . . . . 16 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑓:π‘ βŸΆβ„+)
91 simprl 768 . . . . . . . . . . . . . . . 16 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑧 ∈ 𝑠)
9290, 91ffvelcdmd 7081 . . . . . . . . . . . . . . 15 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (π‘“β€˜π‘§) ∈ ℝ+)
9392rpred 13022 . . . . . . . . . . . . . 14 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (π‘“β€˜π‘§) ∈ ℝ)
9486, 43syl 17 . . . . . . . . . . . . . . 15 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (0(abs ∘ βˆ’ )π‘₯) = (absβ€˜π‘₯))
95 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 = 𝑧 β†’ 𝑒 = 𝑧)
96 fveq2 6885 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 = 𝑧 β†’ (π‘“β€˜π‘’) = (π‘“β€˜π‘§))
9796oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . 22 (𝑒 = 𝑧 β†’ (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) = (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)))
9897ineq1d 4206 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 = 𝑧 β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)) ∩ 𝑋))
9995, 98eqeq12d 2742 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = 𝑧 β†’ (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋) ↔ 𝑧 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)) ∩ 𝑋)))
100 simprr 770 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))
101100ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))
10299, 101, 91rspcdva 3607 . . . . . . . . . . . . . . . . . . 19 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑧 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)) ∩ 𝑋))
10385, 102eleqtrd 2829 . . . . . . . . . . . . . . . . . 18 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ π‘₯ ∈ ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)) ∩ 𝑋))
104103elin1d 4193 . . . . . . . . . . . . . . . . 17 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)))
105 0cnd 11211 . . . . . . . . . . . . . . . . . 18 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ 0 ∈ β„‚)
10692rpxrd 13023 . . . . . . . . . . . . . . . . . 18 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (π‘“β€˜π‘§) ∈ ℝ*)
107 elbl 24249 . . . . . . . . . . . . . . . . . 18 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 0 ∈ β„‚ ∧ (π‘“β€˜π‘§) ∈ ℝ*) β†’ (π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)) ↔ (π‘₯ ∈ β„‚ ∧ (0(abs ∘ βˆ’ )π‘₯) < (π‘“β€˜π‘§))))
10822, 105, 106, 107mp3an2i 1462 . . . . . . . . . . . . . . . . 17 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)) ↔ (π‘₯ ∈ β„‚ ∧ (0(abs ∘ βˆ’ )π‘₯) < (π‘“β€˜π‘§))))
109104, 108mpbid 231 . . . . . . . . . . . . . . . 16 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (π‘₯ ∈ β„‚ ∧ (0(abs ∘ βˆ’ )π‘₯) < (π‘“β€˜π‘§)))
110109simprd 495 . . . . . . . . . . . . . . 15 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (0(abs ∘ βˆ’ )π‘₯) < (π‘“β€˜π‘§))
11194, 110eqbrtrrd 5165 . . . . . . . . . . . . . 14 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (absβ€˜π‘₯) < (π‘“β€˜π‘§))
11296breq1d 5151 . . . . . . . . . . . . . . 15 (𝑒 = 𝑧 β†’ ((π‘“β€˜π‘’) ≀ π‘Ÿ ↔ (π‘“β€˜π‘§) ≀ π‘Ÿ))
113 simplrr 775 . . . . . . . . . . . . . . 15 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)
114112, 113, 91rspcdva 3607 . . . . . . . . . . . . . 14 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (π‘“β€˜π‘§) ≀ π‘Ÿ)
11587, 93, 88, 111, 114ltletrd 11378 . . . . . . . . . . . . 13 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (absβ€˜π‘₯) < π‘Ÿ)
11687, 88, 115ltled 11366 . . . . . . . . . . . 12 (((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) ∧ (𝑧 ∈ 𝑠 ∧ π‘₯ ∈ 𝑧)) β†’ (absβ€˜π‘₯) ≀ π‘Ÿ)
117116rexlimdvaa 3150 . . . . . . . . . . 11 ((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) β†’ (βˆƒπ‘§ ∈ 𝑠 π‘₯ ∈ 𝑧 β†’ (absβ€˜π‘₯) ≀ π‘Ÿ))
11878, 117sylbid 239 . . . . . . . . . 10 ((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) β†’ (π‘₯ ∈ 𝑋 β†’ (absβ€˜π‘₯) ≀ π‘Ÿ))
119118ralrimiv 3139 . . . . . . . . 9 ((((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)) β†’ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)
120 simpllr 773 . . . . . . . . . . 11 (((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ 𝑠 ∈ (𝒫 𝑇 ∩ Fin))
121120elin2d 4194 . . . . . . . . . 10 (((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ 𝑠 ∈ Fin)
122 ffvelcdm 7077 . . . . . . . . . . . . 13 ((𝑓:π‘ βŸΆβ„+ ∧ 𝑒 ∈ 𝑠) β†’ (π‘“β€˜π‘’) ∈ ℝ+)
123122rpred 13022 . . . . . . . . . . . 12 ((𝑓:π‘ βŸΆβ„+ ∧ 𝑒 ∈ 𝑠) β†’ (π‘“β€˜π‘’) ∈ ℝ)
124123ralrimiva 3140 . . . . . . . . . . 11 (𝑓:π‘ βŸΆβ„+ β†’ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ∈ ℝ)
125124ad2antrl 725 . . . . . . . . . 10 (((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ∈ ℝ)
126 fimaxre3 12164 . . . . . . . . . 10 ((𝑠 ∈ Fin ∧ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ∈ ℝ) β†’ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)
127121, 125, 126syl2anc 583 . . . . . . . . 9 (((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ≀ π‘Ÿ)
128119, 127reximddv 3165 . . . . . . . 8 (((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) ∧ (𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)
129128ex 412 . . . . . . 7 ((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) β†’ ((𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ))
130129exlimdv 1928 . . . . . 6 ((((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ βˆͺ 𝑇 = βˆͺ 𝑠) β†’ (βˆƒπ‘“(𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ))
131130expimpd 453 . . . . 5 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) β†’ ((βˆͺ 𝑇 = βˆͺ 𝑠 ∧ βˆƒπ‘“(𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ))
132131rexlimdva 3149 . . . 4 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ (βˆƒπ‘  ∈ (𝒫 𝑇 ∩ Fin)(βˆͺ 𝑇 = βˆͺ 𝑠 ∧ βˆƒπ‘“(𝑓:π‘ βŸΆβ„+ ∧ βˆ€π‘’ ∈ 𝑠 𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))) β†’ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ))
13372, 132mpd 15 . . 3 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)
13410, 133jca 511 . 2 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ (𝑋 ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ))
135 eqid 2726 . . . . . 6 (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i Β· 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i Β· 𝑧)))
136 eqid 2726 . . . . . 6 ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i Β· 𝑧))) β€œ ((-π‘Ÿ[,]π‘Ÿ) Γ— (-π‘Ÿ[,]π‘Ÿ))) = ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i Β· 𝑧))) β€œ ((-π‘Ÿ[,]π‘Ÿ) Γ— (-π‘Ÿ[,]π‘Ÿ)))
1371, 4, 135, 136cnheiborlem 24835 . . . . 5 ((𝑋 ∈ (Clsdβ€˜π½) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)) β†’ 𝑇 ∈ Comp)
138137rexlimdvaa 3150 . . . 4 (𝑋 ∈ (Clsdβ€˜π½) β†’ (βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ β†’ 𝑇 ∈ Comp))
139138imp 406 . . 3 ((𝑋 ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ) β†’ 𝑇 ∈ Comp)
140139adantl 481 . 2 ((𝑋 βŠ† β„‚ ∧ (𝑋 ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)) β†’ 𝑇 ∈ Comp)
141134, 140impbida 798 1 (𝑋 βŠ† β„‚ β†’ (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  βˆͺ cuni 4902   class class class wbr 5141   Γ— cxp 5667   β€œ cima 5672   ∘ ccom 5673  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8941  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113  ici 11114   + caddc 11115   Β· cmul 11117  β„*cxr 11251   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448  -cneg 11449  β„+crp 12980  [,]cicc 13333  abscabs 15187   β†Ύt crest 17375  TopOpenctopn 17376  βˆžMetcxmet 21225  ballcbl 21227  β„‚fldccnfld 21240  Topctop 22750  Clsdccld 22875  Hauscha 23167  Compccmp 23245
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-ioo 13334  df-icc 13337  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-hom 17230  df-cco 17231  df-rest 17377  df-topn 17378  df-0g 17396  df-gsum 17397  df-topgen 17398  df-pt 17399  df-prds 17402  df-xrs 17457  df-qtop 17462  df-imas 17463  df-xps 17465  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-mulg 18996  df-cntz 19233  df-cmn 19702  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-cnfld 21241  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-cld 22878  df-cls 22880  df-cn 23086  df-cnp 23087  df-haus 23174  df-cmp 23246  df-tx 23421  df-hmeo 23614  df-xms 24181  df-ms 24182  df-tms 24183  df-cncf 24753
This theorem is referenced by:  cnllycmp  24837  cncmet  25205  ftalem3  26962
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