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Theorem caubl 25180
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 6887 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑛 β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)))
32sseq1d 4006 . . . . . . . . . . . 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 6887 . . . . . . . . . . . . 13 (π‘Ÿ = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
65sseq1d 4006 . . . . . . . . . . . 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 6887 . . . . . . . . . . . . 13 (π‘Ÿ = (π‘˜ + 1) β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘Ÿ)) = ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))))
98sseq1d 4006 . . . . . . . . . . . 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 3997 . . . . . . . . . . . 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 12901 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ β„• ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘˜ ∈ β„•)
15 fvoveq1 7425 . . . . . . . . . . . . . . . . . . 19 (𝑛 = π‘˜ β†’ (πΉβ€˜(𝑛 + 1)) = (πΉβ€˜(π‘˜ + 1)))
1615fveq2d 6886 . . . . . . . . . . . . . . . . . 18 (𝑛 = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜(𝑛 + 1))) = ((ballβ€˜π·)β€˜(πΉβ€˜(π‘˜ + 1))))
17 2fveq3 6887 . . . . . . . . . . . . . . . . . 18 (𝑛 = π‘˜ β†’ ((ballβ€˜π·)β€˜(πΉβ€˜π‘›)) = ((ballβ€˜π·)β€˜(πΉβ€˜π‘˜)))
1816, 17sseq12d 4008 . . . . . . . . . . . . . . . . 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 3603 . . . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . . 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 12889 . . . . . . . . . 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 744 . . . . . . . 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 5685 . . . . . . . . . . . . . . . 16 Rel (𝑋 Γ— ℝ+)
30 caubl.3 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝐹:β„•βŸΆ(𝑋 Γ— ℝ+))
3130ad3antrrr 727 . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . 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 7078 . . . . . . . . . . . . . . . 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 8019 . . . . . . . . . . . . . . . 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 6886 . . . . . . . . . . . . . 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 7405 . . . . . . . . . . . . . 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 2782 . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . 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 8001 . . . . . . . . . . . . . . 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 8002 . . . . . . . . . . . . . . . 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 13018 . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . 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 13018 . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . 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 12983 . . . . . . . . . . . . . . . . 17 ((2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ+ β†’ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ)
50 rpre 12983 . . . . . . . . . . . . . . . . 17 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ)
51 ltle 11301 . . . . . . . . . . . . . . . . 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 24273 . . . . . . . . . . . . . 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 1378 . . . . . . . . . . . . 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 4013 . . . . . . . . . . . 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 3982 . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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 7078 . . . . . . . . . . . . . 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 8001 . . . . . . . . . . . . . 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 8002 . . . . . . . . . . . . . 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 24263 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 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 8019 . . . . . . . . . . . . . . 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 6886 . . . . . . . . . . . . 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 7405 . . . . . . . . . . . . 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 2782 . . . . . . . . . . . 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 2828 . . . . . . . . . . 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 3968 . . . . . . . . . . 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 24240 . . . . . . . . . . 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 836 . . . . . . . . . 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 3137 . . . . . 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 3160 . . . 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 3159 . . 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 12864 . . 3 β„• = (β„€β‰₯β€˜1)
88 1zzd 12592 . . 3 (πœ‘ β†’ 1 ∈ β„€)
89 fvco3 6981 . . . 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 6981 . . . 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 7999 . . . 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 25152 . 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 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062   βŠ† wss 3941  βŸ¨cop 4627   class class class wbr 5139   Γ— cxp 5665   ∘ ccom 5671  Rel wrel 5672  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  β„cr 11106  1c1 11108   + caddc 11110  β„*cxr 11246   < clt 11247   ≀ cle 11248  β„•cn 12211  β„€cz 12557  β„€β‰₯cuz 12821  β„+crp 12975  βˆžMetcxmet 21219  ballcbl 21221  Cauccau 25125
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12976  df-xneg 13093  df-xadd 13094  df-xmul 13095  df-psmet 21226  df-xmet 21227  df-bl 21229  df-cau 25128
This theorem is referenced by:  bcthlem4  25199  heiborlem9  37191
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