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Theorem caublcls 25058
Description: The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
caubl.2 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
caubl.3 (πœ‘ β†’ 𝐹:β„•βŸΆ(𝑋 Γ— ℝ+))
caubl.4 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))
caublcls.6 𝐽 = (MetOpenβ€˜π·)
Assertion
Ref Expression
caublcls ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ 𝑃 ∈ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
Distinct variable groups:   𝐷,𝑛   𝑛,𝐹   𝑛,𝑋
Allowed substitution hints:   πœ‘(𝑛)   𝐴(𝑛)   𝑃(𝑛)   𝐽(𝑛)

Proof of Theorem caublcls
Dummy variables π‘˜ π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . 2 (β„€β‰₯β€˜π΄) = (β„€β‰₯β€˜π΄)
2 caubl.2 . . . 4 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
323ad2ant1 1132 . . 3 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
4 caublcls.6 . . . 4 𝐽 = (MetOpenβ€˜π·)
54mopntopon 24166 . . 3 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
63, 5syl 17 . 2 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
7 simp3 1137 . . 3 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ 𝐴 ∈ β„•)
87nnzd 12590 . 2 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ 𝐴 ∈ β„€)
9 simp2 1136 . 2 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃)
10 2fveq3 6897 . . . . . . . 8 (π‘Ÿ = 𝐴 β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))
1110sseq1d 4014 . . . . . . 7 (π‘Ÿ = 𝐴 β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) ↔ ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
1211imbi2d 339 . . . . . 6 (π‘Ÿ = 𝐴 β†’ (((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))) ↔ ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))))
13 2fveq3 6897 . . . . . . . 8 (π‘Ÿ = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
1413sseq1d 4014 . . . . . . 7 (π‘Ÿ = π‘˜ β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) ↔ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
1514imbi2d 339 . . . . . 6 (π‘Ÿ = π‘˜ β†’ (((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))) ↔ ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))))
16 2fveq3 6897 . . . . . . . 8 (π‘Ÿ = (π‘˜ + 1) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))))
1716sseq1d 4014 . . . . . . 7 (π‘Ÿ = (π‘˜ + 1) β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) ↔ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
1817imbi2d 339 . . . . . 6 (π‘Ÿ = (π‘˜ + 1) β†’ (((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))) ↔ ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))))
19 ssid 4005 . . . . . . 7 ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))
20192a1i 12 . . . . . 6 (𝐴 ∈ β„€ β†’ ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
21 caubl.4 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))
22 eluznn 12907 . . . . . . . . . . 11 ((𝐴 ∈ β„• ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ π‘˜ ∈ β„•)
23 fvoveq1 7435 . . . . . . . . . . . . . 14 (𝑛 = π‘˜ β†’ (πΉβ€˜(𝑛 + 1)) = (πΉβ€˜(π‘˜ + 1)))
2423fveq2d 6896 . . . . . . . . . . . . 13 (𝑛 = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) = ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))))
25 2fveq3 6897 . . . . . . . . . . . . 13 (𝑛 = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
2624, 25sseq12d 4016 . . . . . . . . . . . 12 (𝑛 = π‘˜ β†’ (((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) ↔ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜))))
2726rspccva 3612 . . . . . . . . . . 11 ((βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) ∧ π‘˜ ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
2821, 22, 27syl2an 595 . . . . . . . . . 10 ((πœ‘ ∧ (𝐴 ∈ β„• ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄))) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
2928anassrs 467 . . . . . . . . 9 (((πœ‘ ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
30 sstr2 3990 . . . . . . . . 9 (((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
3129, 30syl 17 . . . . . . . 8 (((πœ‘ ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
3231expcom 413 . . . . . . 7 (π‘˜ ∈ (β„€β‰₯β€˜π΄) β†’ ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))))
3332a2d 29 . . . . . 6 (π‘˜ ∈ (β„€β‰₯β€˜π΄) β†’ (((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))) β†’ ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))))
3412, 15, 18, 15, 20, 33uzind4 12895 . . . . 5 (π‘˜ ∈ (β„€β‰₯β€˜π΄) β†’ ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
3534impcom 407 . . . 4 (((πœ‘ ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))
36353adantl2 1166 . . 3 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))
373adantr 480 . . . . 5 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
38 simpl1 1190 . . . . . . . 8 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ πœ‘)
39 caubl.3 . . . . . . . 8 (πœ‘ β†’ 𝐹:β„•βŸΆ(𝑋 Γ— ℝ+))
4038, 39syl 17 . . . . . . 7 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ 𝐹:β„•βŸΆ(𝑋 Γ— ℝ+))
41223ad2antl3 1186 . . . . . . 7 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ π‘˜ ∈ β„•)
4240, 41ffvelcdmd 7088 . . . . . 6 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ (πΉβ€˜π‘˜) ∈ (𝑋 Γ— ℝ+))
43 xp1st 8010 . . . . . 6 ((πΉβ€˜π‘˜) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ 𝑋)
4442, 43syl 17 . . . . 5 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ 𝑋)
45 xp2nd 8011 . . . . . 6 ((πΉβ€˜π‘˜) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ+)
4642, 45syl 17 . . . . 5 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ+)
47 blcntr 24140 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜(πΉβ€˜π‘˜)) ∈ 𝑋 ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ+) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ((1st β€˜(πΉβ€˜π‘˜))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘˜))))
4837, 44, 46, 47syl3anc 1370 . . . 4 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ((1st β€˜(πΉβ€˜π‘˜))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘˜))))
49 fvco3 6991 . . . . 5 ((𝐹:β„•βŸΆ(𝑋 Γ— ℝ+) ∧ π‘˜ ∈ β„•) β†’ ((1st ∘ 𝐹)β€˜π‘˜) = (1st β€˜(πΉβ€˜π‘˜)))
5040, 41, 49syl2anc 583 . . . 4 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ ((1st ∘ 𝐹)β€˜π‘˜) = (1st β€˜(πΉβ€˜π‘˜)))
51 1st2nd2 8017 . . . . . . 7 ((πΉβ€˜π‘˜) ∈ (𝑋 Γ— ℝ+) β†’ (πΉβ€˜π‘˜) = ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
5242, 51syl 17 . . . . . 6 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ (πΉβ€˜π‘˜) = ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
5352fveq2d 6896 . . . . 5 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩))
54 df-ov 7415 . . . . 5 ((1st β€˜(πΉβ€˜π‘˜))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘˜))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
5553, 54eqtr4di 2789 . . . 4 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) = ((1st β€˜(πΉβ€˜π‘˜))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘˜))))
5648, 50, 553eltr4d 2847 . . 3 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ ((1st ∘ 𝐹)β€˜π‘˜) ∈ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
5736, 56sseldd 3984 . 2 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ ((1st ∘ 𝐹)β€˜π‘˜) ∈ ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))
5839ffvelcdmda 7087 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ (πΉβ€˜π΄) ∈ (𝑋 Γ— ℝ+))
59583adant2 1130 . . . . . 6 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (πΉβ€˜π΄) ∈ (𝑋 Γ— ℝ+))
60 1st2nd2 8017 . . . . . 6 ((πΉβ€˜π΄) ∈ (𝑋 Γ— ℝ+) β†’ (πΉβ€˜π΄) = ⟨(1st β€˜(πΉβ€˜π΄)), (2nd β€˜(πΉβ€˜π΄))⟩)
6159, 60syl 17 . . . . 5 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (πΉβ€˜π΄) = ⟨(1st β€˜(πΉβ€˜π΄)), (2nd β€˜(πΉβ€˜π΄))⟩)
6261fveq2d 6896 . . . 4 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(πΉβ€˜π΄)), (2nd β€˜(πΉβ€˜π΄))⟩))
63 df-ov 7415 . . . 4 ((1st β€˜(πΉβ€˜π΄))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π΄))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(πΉβ€˜π΄)), (2nd β€˜(πΉβ€˜π΄))⟩)
6462, 63eqtr4di 2789 . . 3 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) = ((1st β€˜(πΉβ€˜π΄))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π΄))))
65 xp1st 8010 . . . . 5 ((πΉβ€˜π΄) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(πΉβ€˜π΄)) ∈ 𝑋)
6659, 65syl 17 . . . 4 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (1st β€˜(πΉβ€˜π΄)) ∈ 𝑋)
67 xp2nd 8011 . . . . . 6 ((πΉβ€˜π΄) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(πΉβ€˜π΄)) ∈ ℝ+)
6859, 67syl 17 . . . . 5 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (2nd β€˜(πΉβ€˜π΄)) ∈ ℝ+)
6968rpxrd 13022 . . . 4 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (2nd β€˜(πΉβ€˜π΄)) ∈ ℝ*)
70 blssm 24145 . . . 4 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜(πΉβ€˜π΄)) ∈ 𝑋 ∧ (2nd β€˜(πΉβ€˜π΄)) ∈ ℝ*) β†’ ((1st β€˜(πΉβ€˜π΄))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π΄))) βŠ† 𝑋)
713, 66, 69, 70syl3anc 1370 . . 3 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ ((1st β€˜(πΉβ€˜π΄))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π΄))) βŠ† 𝑋)
7264, 71eqsstrd 4021 . 2 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π΄)) βŠ† 𝑋)
731, 6, 8, 9, 57, 72lmcls 23027 1 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ 𝑃 ∈ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(πΉβ€˜π΄))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   βŠ† wss 3949  βŸ¨cop 4635   class class class wbr 5149   Γ— cxp 5675   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977  1c1 11114   + caddc 11116  β„*cxr 11252  β„•cn 12217  β„€cz 12563  β„€β‰₯cuz 12827  β„+crp 12979  βˆžMetcxmet 21130  ballcbl 21132  MetOpencmopn 21135  TopOnctopon 22633  clsccl 22743  β‡π‘‘clm 22951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9440  df-inf 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-n0 12478  df-z 12564  df-uz 12828  df-q 12938  df-rp 12980  df-xneg 13097  df-xadd 13098  df-xmul 13099  df-topgen 17394  df-psmet 21137  df-xmet 21138  df-bl 21140  df-mopn 21141  df-top 22617  df-topon 22634  df-bases 22670  df-cld 22744  df-ntr 22745  df-cls 22746  df-lm 22954
This theorem is referenced by:  bcthlem3  25075  heiborlem8  36990
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