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Theorem caubl 25249
Description: Sufficient condition to ensure a sequence of nested balls is Cauchy. (Contributed by Mario Carneiro, 18-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‘›)))
caubl.5 (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘› ∈ β„• (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)
Assertion
Ref Expression
caubl (πœ‘ β†’ (1st ∘ 𝐹) ∈ (Cauβ€˜π·))
Distinct variable groups:   𝑛,π‘Ÿ,𝐷   𝑛,𝐹,π‘Ÿ   πœ‘,π‘Ÿ   𝑛,𝑋,π‘Ÿ   πœ‘,𝑛

Proof of Theorem caubl
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 caubl.5 . . 3 (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘› ∈ β„• (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)
2 2fveq3 6902 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑛 β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))
32sseq1d 4011 . . . . . . . . . . . 12 (π‘Ÿ = 𝑛 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‘›))))
43imbi2d 340 . . . . . . . . . . 11 (π‘Ÿ = 𝑛 β†’ (((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))) ↔ ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))))
5 2fveq3 6902 . . . . . . . . . . . . 13 (π‘Ÿ = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
65sseq1d 4011 . . . . . . . . . . . 12 (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‘›))))
76imbi2d 340 . . . . . . . . . . 11 (π‘Ÿ = π‘˜ β†’ (((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))) ↔ ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))))
8 2fveq3 6902 . . . . . . . . . . . . 13 (π‘Ÿ = (π‘˜ + 1) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))))
98sseq1d 4011 . . . . . . . . . . . 12 (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‘›))))
109imbi2d 340 . . . . . . . . . . 11 (π‘Ÿ = (π‘˜ + 1) β†’ (((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))) ↔ ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))))
11 ssid 4002 . . . . . . . . . . . 12 ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))
12112a1i 12 . . . . . . . . . . 11 (𝑛 ∈ β„€ β†’ ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))))
13 caubl.4 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))
14 eluznn 12933 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ β„• ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘˜ ∈ β„•)
15 fvoveq1 7443 . . . . . . . . . . . . . . . . . . 19 (𝑛 = π‘˜ β†’ (πΉβ€˜(𝑛 + 1)) = (πΉβ€˜(π‘˜ + 1)))
1615fveq2d 6901 . . . . . . . . . . . . . . . . . 18 (𝑛 = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) = ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))))
17 2fveq3 6902 . . . . . . . . . . . . . . . . . 18 (𝑛 = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
1816, 17sseq12d 4013 . . . . . . . . . . . . . . . . 17 (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‘˜))))
1918rspccva 3608 . . . . . . . . . . . . . . . 16 ((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‘˜)))
2013, 14, 19syl2an 595 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
2120anassrs 467 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
22 sstr2 3987 . . . . . . . . . . . . . 14 (((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‘›))))
2321, 22syl 17 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑛 ∈ 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‘›))))
2423expcom 413 . . . . . . . . . . . 12 (π‘˜ ∈ (β„€β‰₯β€˜π‘›) β†’ ((πœ‘ ∧ 𝑛 ∈ 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‘›)))))
2524a2d 29 . . . . . . . . . . 11 (π‘˜ ∈ (β„€β‰₯β€˜π‘›) β†’ (((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))) β†’ ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))))
264, 7, 10, 7, 12, 25uzind4 12921 . . . . . . . . . 10 (π‘˜ ∈ (β„€β‰₯β€˜π‘›) β†’ ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))))
2726com12 32 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘˜ ∈ (β„€β‰₯β€˜π‘›) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))))
2827ad2ant2r 746 . . . . . . . 8 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) β†’ (π‘˜ ∈ (β„€β‰₯β€˜π‘›) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›))))
29 relxp 5696 . . . . . . . . . . . . . . . 16 Rel (𝑋 Γ— ℝ+)
30 caubl.3 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝐹:β„•βŸΆ(𝑋 Γ— ℝ+))
3130ad3antrrr 729 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ 𝐹:β„•βŸΆ(𝑋 Γ— ℝ+))
32 simplrl 776 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ 𝑛 ∈ β„•)
3331, 32ffvelcdmd 7095 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (πΉβ€˜π‘›) ∈ (𝑋 Γ— ℝ+))
34 1st2nd 8043 . . . . . . . . . . . . . . . 16 ((Rel (𝑋 Γ— ℝ+) ∧ (πΉβ€˜π‘›) ∈ (𝑋 Γ— ℝ+)) β†’ (πΉβ€˜π‘›) = ⟨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩)
3529, 33, 34sylancr 586 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (πΉβ€˜π‘›) = ⟨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩)
3635fveq2d 6901 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ 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‘›))⟩))
37 df-ov 7423 . . . . . . . . . . . . . 14 ((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‘›))⟩)
3836, 37eqtr4di 2786 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) = ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘›))))
39 caubl.2 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
4039ad3antrrr 729 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
41 xp1st 8025 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘›) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(πΉβ€˜π‘›)) ∈ 𝑋)
4233, 41syl 17 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (1st β€˜(πΉβ€˜π‘›)) ∈ 𝑋)
43 xp2nd 8026 . . . . . . . . . . . . . . . 16 ((πΉβ€˜π‘›) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ+)
4433, 43syl 17 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ+)
4544rpxrd 13050 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ*)
46 simpllr 775 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘Ÿ ∈ ℝ+)
4746rpxrd 13050 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘Ÿ ∈ ℝ*)
48 simplrr 777 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)
49 rpre 13015 . . . . . . . . . . . . . . . . 17 ((2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ+ β†’ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ)
50 rpre 13015 . . . . . . . . . . . . . . . . 17 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ)
51 ltle 11333 . . . . . . . . . . . . . . . . 17 (((2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ ∧ π‘Ÿ ∈ ℝ) β†’ ((2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ β†’ (2nd β€˜(πΉβ€˜π‘›)) ≀ π‘Ÿ))
5249, 50, 51syl2an 595 . . . . . . . . . . . . . . . 16 (((2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ+ ∧ π‘Ÿ ∈ ℝ+) β†’ ((2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ β†’ (2nd β€˜(πΉβ€˜π‘›)) ≀ π‘Ÿ))
5344, 46, 52syl2anc 583 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ ((2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ β†’ (2nd β€˜(πΉβ€˜π‘›)) ≀ π‘Ÿ))
5448, 53mpd 15 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (2nd β€˜(πΉβ€˜π‘›)) ≀ π‘Ÿ)
55 ssbl 24342 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜(πΉβ€˜π‘›)) ∈ 𝑋) ∧ ((2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ* ∧ π‘Ÿ ∈ ℝ*) ∧ (2nd β€˜(πΉβ€˜π‘›)) ≀ π‘Ÿ) β†’ ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘›))) βŠ† ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ))
5640, 42, 45, 47, 54, 55syl221anc 1379 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘›))) βŠ† ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ))
5738, 56eqsstrd 4018 . . . . . . . . . . . 12 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) βŠ† ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ))
58 sstr2 3987 . . . . . . . . . . . 12 (((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) βŠ† ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ)))
5957, 58syl5com 31 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ 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‘˜)) βŠ† ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ)))
60 simprl 770 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) β†’ 𝑛 ∈ β„•)
6160, 14sylan 579 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘˜ ∈ β„•)
6231, 61ffvelcdmd 7095 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (πΉβ€˜π‘˜) ∈ (𝑋 Γ— ℝ+))
63 xp1st 8025 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘˜) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ 𝑋)
6462, 63syl 17 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ 𝑋)
65 xp2nd 8026 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘˜) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ+)
6662, 65syl 17 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ+)
67 blcntr 24332 . . . . . . . . . . . . 13 ((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‘˜))))
6840, 64, 66, 67syl3anc 1369 . . . . . . . . . . . 12 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ((1st β€˜(πΉβ€˜π‘˜))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘˜))))
69 1st2nd 8043 . . . . . . . . . . . . . . 15 ((Rel (𝑋 Γ— ℝ+) ∧ (πΉβ€˜π‘˜) ∈ (𝑋 Γ— ℝ+)) β†’ (πΉβ€˜π‘˜) = ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
7029, 62, 69sylancr 586 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (πΉβ€˜π‘˜) = ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
7170fveq2d 6901 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ 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‘˜))⟩))
72 df-ov 7423 . . . . . . . . . . . . 13 ((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‘˜))⟩)
7371, 72eqtr4di 2786 . . . . . . . . . . . 12 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) = ((1st β€˜(πΉβ€˜π‘˜))(ballβ€˜π·)(2nd β€˜(πΉβ€˜π‘˜))))
7468, 73eleqtrrd 2832 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
75 ssel 3973 . . . . . . . . . . 11 (((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ) β†’ ((1st β€˜(πΉβ€˜π‘˜)) ∈ ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ)))
7659, 74, 75syl6ci 71 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ)))
77 elbl2 24309 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ*) ∧ ((1st β€˜(πΉβ€˜π‘›)) ∈ 𝑋 ∧ (1st β€˜(πΉβ€˜π‘˜)) ∈ 𝑋)) β†’ ((1st β€˜(πΉβ€˜π‘˜)) ∈ ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ) ↔ ((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ))
7840, 47, 42, 64, 77syl22anc 838 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ ((1st β€˜(πΉβ€˜π‘˜)) ∈ ((1st β€˜(πΉβ€˜π‘›))(ballβ€˜π·)π‘Ÿ) ↔ ((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ))
7976, 78sylibd 238 . . . . . . . . 9 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) β†’ ((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ))
8079ex 412 . . . . . . . 8 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) β†’ (π‘˜ ∈ (β„€β‰₯β€˜π‘›) β†’ (((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)) βŠ† ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) β†’ ((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ)))
8128, 80mpdd 43 . . . . . . 7 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) β†’ (π‘˜ ∈ (β„€β‰₯β€˜π‘›) β†’ ((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ))
8281ralrimiv 3142 . . . . . 6 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ (𝑛 ∈ β„• ∧ (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ)) β†’ βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ)
8382expr 456 . . . . 5 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑛 ∈ β„•) β†’ ((2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ β†’ βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ))
8483reximdva 3165 . . . 4 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ (βˆƒπ‘› ∈ β„• (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ β†’ βˆƒπ‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ))
8584ralimdva 3164 . . 3 (πœ‘ β†’ (βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘› ∈ β„• (2nd β€˜(πΉβ€˜π‘›)) < π‘Ÿ β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ))
861, 85mpd 15 . 2 (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ)
87 nnuz 12896 . . 3 β„• = (β„€β‰₯β€˜1)
88 1zzd 12624 . . 3 (πœ‘ β†’ 1 ∈ β„€)
89 fvco3 6997 . . . 4 ((𝐹:β„•βŸΆ(𝑋 Γ— ℝ+) ∧ π‘˜ ∈ β„•) β†’ ((1st ∘ 𝐹)β€˜π‘˜) = (1st β€˜(πΉβ€˜π‘˜)))
9030, 89sylan 579 . . 3 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ ((1st ∘ 𝐹)β€˜π‘˜) = (1st β€˜(πΉβ€˜π‘˜)))
91 fvco3 6997 . . . 4 ((𝐹:β„•βŸΆ(𝑋 Γ— ℝ+) ∧ 𝑛 ∈ β„•) β†’ ((1st ∘ 𝐹)β€˜π‘›) = (1st β€˜(πΉβ€˜π‘›)))
9230, 91sylan 579 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((1st ∘ 𝐹)β€˜π‘›) = (1st β€˜(πΉβ€˜π‘›)))
93 1stcof 8023 . . . 4 (𝐹:β„•βŸΆ(𝑋 Γ— ℝ+) β†’ (1st ∘ 𝐹):β„•βŸΆπ‘‹)
9430, 93syl 17 . . 3 (πœ‘ β†’ (1st ∘ 𝐹):β„•βŸΆπ‘‹)
9587, 39, 88, 90, 92, 94iscauf 25221 . 2 (πœ‘ β†’ ((1st ∘ 𝐹) ∈ (Cauβ€˜π·) ↔ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((1st β€˜(πΉβ€˜π‘›))𝐷(1st β€˜(πΉβ€˜π‘˜))) < π‘Ÿ))
9686, 95mpbird 257 1 (πœ‘ β†’ (1st ∘ 𝐹) ∈ (Cauβ€˜π·))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  βˆƒwrex 3067   βŠ† wss 3947  βŸ¨cop 4635   class class class wbr 5148   Γ— cxp 5676   ∘ ccom 5682  Rel wrel 5683  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  β„cr 11138  1c1 11140   + caddc 11142  β„*cxr 11278   < clt 11279   ≀ cle 11280  β„•cn 12243  β„€cz 12589  β„€β‰₯cuz 12853  β„+crp 13007  βˆžMetcxmet 21264  ballcbl 21266  Cauccau 25194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-n0 12504  df-z 12590  df-uz 12854  df-rp 13008  df-xneg 13125  df-xadd 13126  df-xmul 13127  df-psmet 21271  df-xmet 21272  df-bl 21274  df-cau 25197
This theorem is referenced by:  bcthlem4  25268  heiborlem9  37292
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