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Theorem caublcls 24833
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 2732 . 2 (β„€β‰₯β€˜π΄) = (β„€β‰₯β€˜π΄)
2 caubl.2 . . . 4 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
323ad2ant1 1133 . . 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 23952 . . 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 1138 . . 3 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ 𝐴 ∈ β„•)
87nnzd 12587 . 2 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ 𝐴 ∈ β„€)
9 simp2 1137 . 2 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃)
10 2fveq3 6896 . . . . . . . 8 (π‘Ÿ = 𝐴 β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜π΄)))
1110sseq1d 4013 . . . . . . 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 340 . . . . . 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 6896 . . . . . . . 8 (π‘Ÿ = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
1413sseq1d 4013 . . . . . . 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 340 . . . . . 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 6896 . . . . . . . 8 (π‘Ÿ = (π‘˜ + 1) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))))
1716sseq1d 4013 . . . . . . 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 340 . . . . . 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 4004 . . . . . . 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 12904 . . . . . . . . . . 11 ((𝐴 ∈ β„• ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ π‘˜ ∈ β„•)
23 fvoveq1 7434 . . . . . . . . . . . . . 14 (𝑛 = π‘˜ β†’ (πΉβ€˜(𝑛 + 1)) = (πΉβ€˜(π‘˜ + 1)))
2423fveq2d 6895 . . . . . . . . . . . . 13 (𝑛 = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) = ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))))
25 2fveq3 6896 . . . . . . . . . . . . 13 (𝑛 = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
2624, 25sseq12d 4015 . . . . . . . . . . . 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 3611 . . . . . . . . . . 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 596 . . . . . . . . . 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 468 . . . . . . . . 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 3989 . . . . . . . . 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 414 . . . . . . 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 12892 . . . . 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 408 . . . 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 1167 . . 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 481 . . . . 5 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
38 simpl1 1191 . . . . . . . 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 1187 . . . . . . 7 (((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π΄)) β†’ π‘˜ ∈ β„•)
4240, 41ffvelcdmd 7087 . . . . . 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 8009 . . . . . 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 8010 . . . . . 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 23926 . . . . 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 1371 . . . 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 6990 . . . . 5 ((𝐹:β„•βŸΆ(𝑋 Γ— ℝ+) ∧ π‘˜ ∈ β„•) β†’ ((1st ∘ 𝐹)β€˜π‘˜) = (1st β€˜(πΉβ€˜π‘˜)))
5040, 41, 49syl2anc 584 . . . 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 8016 . . . . . . 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 6895 . . . . 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 7414 . . . . 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 2790 . . . 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 2848 . . 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 3983 . 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 7086 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ (πΉβ€˜π΄) ∈ (𝑋 Γ— ℝ+))
59583adant2 1131 . . . . . 6 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (πΉβ€˜π΄) ∈ (𝑋 Γ— ℝ+))
60 1st2nd2 8016 . . . . . 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 6895 . . . 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 7414 . . . 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 2790 . . 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 8009 . . . . 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 8010 . . . . . 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 13019 . . . 4 ((πœ‘ ∧ (1st ∘ 𝐹)(β‡π‘‘β€˜π½)𝑃 ∧ 𝐴 ∈ β„•) β†’ (2nd β€˜(πΉβ€˜π΄)) ∈ ℝ*)
70 blssm 23931 . . . 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 1371 . . 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 4020 . 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 22813 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  βŸ¨cop 4634   class class class wbr 5148   Γ— cxp 5674   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  1c1 11113   + caddc 11115  β„*cxr 11249  β„•cn 12214  β„€cz 12560  β„€β‰₯cuz 12824  β„+crp 12976  βˆžMetcxmet 20935  ballcbl 20937  MetOpencmopn 20940  TopOnctopon 22419  clsccl 22529  β‡π‘‘clm 22737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  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
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-bl 20945  df-mopn 20946  df-top 22403  df-topon 22420  df-bases 22456  df-cld 22530  df-ntr 22531  df-cls 22532  df-lm 22740
This theorem is referenced by:  bcthlem3  24850  heiborlem8  36772
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