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Theorem cnheibor 24899
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 24719 . . . 4 𝐽 ∈ Haus
3 simpl 481 . . . 4 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 βŠ† β„‚)
4 cnheibor.3 . . . . 5 𝑇 = (𝐽 β†Ύt 𝑋)
5 simpr 483 . . . . 5 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑇 ∈ Comp)
64, 5eqeltrrid 2830 . . . 4 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ (𝐽 β†Ύt 𝑋) ∈ Comp)
71cnfldtopon 24717 . . . . . 6 𝐽 ∈ (TopOnβ€˜β„‚)
87toponunii 22836 . . . . 5 β„‚ = βˆͺ 𝐽
98hauscmp 23329 . . . 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 24718 . . . . . . . . . . 11 𝐽 ∈ Top
128restuni 23084 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 βŠ† β„‚) β†’ 𝑋 = βˆͺ (𝐽 β†Ύt 𝑋))
1311, 3, 12sylancr 585 . . . . . . . . . 10 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 = βˆͺ (𝐽 β†Ύt 𝑋))
144unieqi 4915 . . . . . . . . . 10 βˆͺ 𝑇 = βˆͺ (𝐽 β†Ύt 𝑋)
1513, 14eqtr4di 2783 . . . . . . . . 9 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 = βˆͺ 𝑇)
1615eleq2d 2811 . . . . . . . 8 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝑇))
1716biimpar 476 . . . . . . 7 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ βˆͺ 𝑇) β†’ π‘₯ ∈ 𝑋)
18 cnex 11219 . . . . . . . . . . . 12 β„‚ ∈ V
19 ssexg 5318 . . . . . . . . . . . 12 ((𝑋 βŠ† β„‚ ∧ β„‚ ∈ V) β†’ 𝑋 ∈ V)
203, 18, 19sylancl 584 . . . . . . . . . . 11 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ 𝑋 ∈ V)
2120adantr 479 . . . . . . . . . 10 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 ∈ V)
22 cnxmet 24707 . . . . . . . . . . 11 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
23 0cnd 11237 . . . . . . . . . . 11 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ 0 ∈ β„‚)
243sselda 3972 . . . . . . . . . . . . . 14 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ β„‚)
2524abscld 15415 . . . . . . . . . . . . 13 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π‘₯) ∈ ℝ)
26 peano2re 11417 . . . . . . . . . . . . 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 11294 . . . . . . . . . . 11 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ ((absβ€˜π‘₯) + 1) ∈ ℝ*)
291cnfldtopn 24716 . . . . . . . . . . . 12 𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))
3029blopn 24427 . . . . . . . . . . 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 17409 . . . . . . . . . 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 2836 . . . . . . . 8 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) ∈ 𝑇)
35 0cn 11236 . . . . . . . . . . . . . 14 0 ∈ β„‚
36 eqid 2725 . . . . . . . . . . . . . . 15 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
3736cnmetdval 24705 . . . . . . . . . . . . . 14 ((0 ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ (0(abs ∘ βˆ’ )π‘₯) = (absβ€˜(0 βˆ’ π‘₯)))
3835, 37mpan 688 . . . . . . . . . . . . 13 (π‘₯ ∈ β„‚ β†’ (0(abs ∘ βˆ’ )π‘₯) = (absβ€˜(0 βˆ’ π‘₯)))
39 df-neg 11477 . . . . . . . . . . . . . . 15 -π‘₯ = (0 βˆ’ π‘₯)
4039fveq2i 6895 . . . . . . . . . . . . . 14 (absβ€˜-π‘₯) = (absβ€˜(0 βˆ’ π‘₯))
41 absneg 15256 . . . . . . . . . . . . . 14 (π‘₯ ∈ β„‚ β†’ (absβ€˜-π‘₯) = (absβ€˜π‘₯))
4240, 41eqtr3id 2779 . . . . . . . . . . . . 13 (π‘₯ ∈ β„‚ β†’ (absβ€˜(0 βˆ’ π‘₯)) = (absβ€˜π‘₯))
4338, 42eqtrd 2765 . . . . . . . . . . . 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 12174 . . . . . . . . . . 11 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π‘₯) < ((absβ€˜π‘₯) + 1))
4644, 45eqbrtrd 5165 . . . . . . . . . 10 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ (0(abs ∘ βˆ’ )π‘₯) < ((absβ€˜π‘₯) + 1))
47 elbl 24312 . . . . . . . . . . 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 711 . . . . . . . . 9 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)))
50 simpr 483 . . . . . . . . 9 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
5149, 50elind 4188 . . . . . . . 8 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋))
5224absge0d 15423 . . . . . . . . . 10 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ 0 ≀ (absβ€˜π‘₯))
5325, 52ge0p1rpd 13078 . . . . . . . . 9 (((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) ∧ π‘₯ ∈ 𝑋) β†’ ((absβ€˜π‘₯) + 1) ∈ ℝ+)
54 eqid 2725 . . . . . . . . 9 ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋)
55 oveq2 7424 . . . . . . . . . . 11 (π‘Ÿ = ((absβ€˜π‘₯) + 1) β†’ (0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) = (0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)))
5655ineq1d 4205 . . . . . . . . . 10 (π‘Ÿ = ((absβ€˜π‘₯) + 1) β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋))
5756rspceeqv 3623 . . . . . . . . 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 584 . . . . . . . 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 2814 . . . . . . . . . 10 (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋) β†’ (π‘₯ ∈ 𝑒 ↔ π‘₯ ∈ ((0(ballβ€˜(abs ∘ βˆ’ ))((absβ€˜π‘₯) + 1)) ∩ 𝑋)))
60 eqeq1 2729 . . . . . . . . . . 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 3169 . . . . . . . . . 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 3601 . . . . . . . 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 835 . . . . . . 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 589 . . . . . 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 3136 . . . . 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 2725 . . . . . 6 βˆͺ 𝑇 = βˆͺ 𝑇
68 oveq2 7424 . . . . . . . 8 (π‘Ÿ = (π‘“β€˜π‘’) β†’ (0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) = (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)))
6968ineq1d 4205 . . . . . . 7 (π‘Ÿ = (π‘“β€˜π‘’) β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋))
7069eqeq2d 2736 . . . . . 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 23313 . . . . 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 582 . . . 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 730 . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . 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 2811 . . . . . . . . . . . 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 4907 . . . . . . . . . . . 12 (π‘₯ ∈ βˆͺ 𝑠 ↔ βˆƒπ‘§ ∈ 𝑠 π‘₯ ∈ 𝑧)
7876, 77bitrdi 286 . . . . . . . . . . 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 4935 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ 𝑠 β†’ 𝑧 βŠ† βˆͺ 𝑠)
8079ad2antrl 726 . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . 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 4014 . . . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 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 3973 . . . . . . . . . . . . . 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 15415 . . . . . . . . . . . . 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 775 . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 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 7090 . . . . . . . . . . . . . . 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 13048 . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 = 𝑧 β†’ (π‘“β€˜π‘’) = (π‘“β€˜π‘§))
9796oveq2d 7432 . . . . . . . . . . . . . . . . . . . . . 22 (𝑒 = 𝑧 β†’ (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) = (0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)))
9897ineq1d 4205 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 = 𝑧 β†’ ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋) = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)) ∩ 𝑋))
9995, 98eqeq12d 2741 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = 𝑧 β†’ (𝑒 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘’)) ∩ 𝑋) ↔ 𝑧 = ((0(ballβ€˜(abs ∘ βˆ’ ))(π‘“β€˜π‘§)) ∩ 𝑋)))
100 simprr 771 . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . 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 3602 . . . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . . . . 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 4192 . . . . . . . . . . . . . . . . 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 11237 . . . . . . . . . . . . . . . . . 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 13049 . . . . . . . . . . . . . . . . . 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 24312 . . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . 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 5167 . . . . . . . . . . . . . 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 5153 . . . . . . . . . . . . . . 15 (𝑒 = 𝑧 β†’ ((π‘“β€˜π‘’) ≀ π‘Ÿ ↔ (π‘“β€˜π‘§) ≀ π‘Ÿ))
113 simplrr 776 . . . . . . . . . . . . . . 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 3602 . . . . . . . . . . . . . 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 11404 . . . . . . . . . . . . 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 11392 . . . . . . . . . . . 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 3146 . . . . . . . . . . 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 3135 . . . . . . . . 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 774 . . . . . . . . . . 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 4193 . . . . . . . . . 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 7086 . . . . . . . . . . . . 13 ((𝑓:π‘ βŸΆβ„+ ∧ 𝑒 ∈ 𝑠) β†’ (π‘“β€˜π‘’) ∈ ℝ+)
123122rpred 13048 . . . . . . . . . . . 12 ((𝑓:π‘ βŸΆβ„+ ∧ 𝑒 ∈ 𝑠) β†’ (π‘“β€˜π‘’) ∈ ℝ)
124123ralrimiva 3136 . . . . . . . . . . 11 (𝑓:π‘ βŸΆβ„+ β†’ βˆ€π‘’ ∈ 𝑠 (π‘“β€˜π‘’) ∈ ℝ)
125124ad2antrl 726 . . . . . . . . . 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 12190 . . . . . . . . . 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 582 . . . . . . . . 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 3161 . . . . . . . 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 411 . . . . . . 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 452 . . . . 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 3145 . . . 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 510 . 2 ((𝑋 βŠ† β„‚ ∧ 𝑇 ∈ Comp) β†’ (𝑋 ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ))
135 eqid 2725 . . . . . 6 (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i Β· 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i Β· 𝑧)))
136 eqid 2725 . . . . . 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 24898 . . . . 5 ((𝑋 ∈ (Clsdβ€˜π½) ∧ (π‘Ÿ ∈ ℝ ∧ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)) β†’ 𝑇 ∈ Comp)
138137rexlimdvaa 3146 . . . 4 (𝑋 ∈ (Clsdβ€˜π½) β†’ (βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ β†’ 𝑇 ∈ Comp))
139138imp 405 . . 3 ((𝑋 ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ) β†’ 𝑇 ∈ Comp)
140139adantl 480 . 2 ((𝑋 βŠ† β„‚ ∧ (𝑋 ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘Ÿ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜π‘₯) ≀ π‘Ÿ)) β†’ 𝑇 ∈ Comp)
141134, 140impbida 799 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 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3051  βˆƒwrex 3060  Vcvv 3463   ∩ cin 3938   βŠ† wss 3939  π’« cpw 4598  βˆͺ cuni 4903   class class class wbr 5143   Γ— cxp 5670   β€œ cima 5675   ∘ ccom 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  Fincfn 8962  β„‚cc 11136  β„cr 11137  0cc0 11138  1c1 11139  ici 11140   + caddc 11141   Β· cmul 11143  β„*cxr 11277   < clt 11278   ≀ cle 11279   βˆ’ cmin 11474  -cneg 11475  β„+crp 13006  [,]cicc 13359  abscabs 15213   β†Ύt crest 17401  TopOpenctopn 17402  βˆžMetcxmet 21268  ballcbl 21270  β„‚fldccnfld 21283  Topctop 22813  Clsdccld 22938  Hauscha 23230  Compccmp 23308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216  ax-addf 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-supp 8164  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8723  df-map 8845  df-ixp 8915  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-fsupp 9386  df-fi 9434  df-sup 9465  df-inf 9466  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-q 12963  df-rp 13007  df-xneg 13124  df-xadd 13125  df-xmul 13126  df-ioo 13360  df-icc 13363  df-fz 13517  df-fzo 13660  df-seq 13999  df-exp 14059  df-hash 14322  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-starv 17247  df-sca 17248  df-vsca 17249  df-ip 17250  df-tset 17251  df-ple 17252  df-ds 17254  df-unif 17255  df-hom 17256  df-cco 17257  df-rest 17403  df-topn 17404  df-0g 17422  df-gsum 17423  df-topgen 17424  df-pt 17425  df-prds 17428  df-xrs 17483  df-qtop 17488  df-imas 17489  df-xps 17491  df-mre 17565  df-mrc 17566  df-acs 17568  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18740  df-mulg 19028  df-cntz 19272  df-cmn 19741  df-psmet 21275  df-xmet 21276  df-met 21277  df-bl 21278  df-mopn 21279  df-cnfld 21284  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22867  df-cld 22941  df-cls 22943  df-cn 23149  df-cnp 23150  df-haus 23237  df-cmp 23309  df-tx 23484  df-hmeo 23677  df-xms 24244  df-ms 24245  df-tms 24246  df-cncf 24816
This theorem is referenced by:  cnllycmp  24900  cncmet  25268  ftalem3  27025
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