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Theorem bcthlem4 24694
Description: Lemma for bcth 24696. Given any open ball (𝐢(ballβ€˜π·)𝑅) as starting point (and in particular, a ball in int(βˆͺ ran 𝑀)), the limit point π‘₯ of the centers of the induced sequence of balls 𝑔 is outside βˆͺ ran 𝑀. Note that a set 𝐴 has empty interior iff every nonempty open set π‘ˆ contains points outside 𝐴, i.e. (π‘ˆ βˆ– 𝐴) β‰  βˆ…. (Contributed by Mario Carneiro, 7-Jan-2014.)
Hypotheses
Ref Expression
bcth.2 𝐽 = (MetOpenβ€˜π·)
bcthlem.4 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
bcthlem.5 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
bcthlem.6 (πœ‘ β†’ 𝑀:β„•βŸΆ(Clsdβ€˜π½))
bcthlem.7 (πœ‘ β†’ 𝑅 ∈ ℝ+)
bcthlem.8 (πœ‘ β†’ 𝐢 ∈ 𝑋)
bcthlem.9 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
bcthlem.10 (πœ‘ β†’ (π‘”β€˜1) = ⟨𝐢, π‘…βŸ©)
bcthlem.11 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
Assertion
Ref Expression
bcthlem4 (πœ‘ β†’ ((𝐢(ballβ€˜π·)𝑅) βˆ– βˆͺ ran 𝑀) β‰  βˆ…)
Distinct variable groups:   π‘˜,π‘Ÿ,π‘₯,𝑧   𝐢,π‘Ÿ,π‘₯   𝑔,π‘˜,π‘Ÿ,π‘₯,𝑧,𝐷   𝑔,𝐹,π‘˜,π‘Ÿ,π‘₯,𝑧   𝑔,𝐽,π‘˜,π‘Ÿ,π‘₯,𝑧   𝑔,𝑀,π‘˜,π‘Ÿ,π‘₯,𝑧   πœ‘,π‘˜,π‘Ÿ,π‘₯,𝑧   π‘₯,𝑅   𝑔,𝑋,π‘˜,π‘Ÿ,π‘₯,𝑧
Allowed substitution hints:   πœ‘(𝑔)   𝐢(𝑧,𝑔,π‘˜)   𝑅(𝑧,𝑔,π‘˜,π‘Ÿ)

Proof of Theorem bcthlem4
Dummy variables 𝑛 π‘š 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcthlem.4 . . . 4 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
2 cmetmet 24653 . . . . . . 7 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
31, 2syl 17 . . . . . 6 (πœ‘ β†’ 𝐷 ∈ (Metβ€˜π‘‹))
4 metxmet 23690 . . . . . 6 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
53, 4syl 17 . . . . 5 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
6 bcthlem.9 . . . . 5 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
7 bcth.2 . . . . . 6 𝐽 = (MetOpenβ€˜π·)
8 bcthlem.5 . . . . . 6 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
9 bcthlem.6 . . . . . 6 (πœ‘ β†’ 𝑀:β„•βŸΆ(Clsdβ€˜π½))
10 bcthlem.7 . . . . . 6 (πœ‘ β†’ 𝑅 ∈ ℝ+)
11 bcthlem.8 . . . . . 6 (πœ‘ β†’ 𝐢 ∈ 𝑋)
12 bcthlem.10 . . . . . 6 (πœ‘ β†’ (π‘”β€˜1) = ⟨𝐢, π‘…βŸ©)
13 bcthlem.11 . . . . . 6 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
147, 1, 8, 9, 10, 11, 6, 12, 13bcthlem2 24692 . . . . 5 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
15 elrp 12918 . . . . . . . . 9 (π‘Ÿ ∈ ℝ+ ↔ (π‘Ÿ ∈ ℝ ∧ 0 < π‘Ÿ))
16 nnrecl 12412 . . . . . . . . 9 ((π‘Ÿ ∈ ℝ ∧ 0 < π‘Ÿ) β†’ βˆƒπ‘š ∈ β„• (1 / π‘š) < π‘Ÿ)
1715, 16sylbi 216 . . . . . . . 8 (π‘Ÿ ∈ ℝ+ β†’ βˆƒπ‘š ∈ β„• (1 / π‘š) < π‘Ÿ)
1817adantl 483 . . . . . . 7 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ βˆƒπ‘š ∈ β„• (1 / π‘š) < π‘Ÿ)
19 peano2nn 12166 . . . . . . . . . 10 (π‘š ∈ β„• β†’ (π‘š + 1) ∈ β„•)
2019adantl 483 . . . . . . . . 9 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (π‘š + 1) ∈ β„•)
21 fvoveq1 7381 . . . . . . . . . . . . . . . 16 (π‘˜ = π‘š β†’ (π‘”β€˜(π‘˜ + 1)) = (π‘”β€˜(π‘š + 1)))
22 id 22 . . . . . . . . . . . . . . . . 17 (π‘˜ = π‘š β†’ π‘˜ = π‘š)
23 fveq2 6843 . . . . . . . . . . . . . . . . 17 (π‘˜ = π‘š β†’ (π‘”β€˜π‘˜) = (π‘”β€˜π‘š))
2422, 23oveq12d 7376 . . . . . . . . . . . . . . . 16 (π‘˜ = π‘š β†’ (π‘˜πΉ(π‘”β€˜π‘˜)) = (π‘šπΉ(π‘”β€˜π‘š)))
2521, 24eleq12d 2832 . . . . . . . . . . . . . . 15 (π‘˜ = π‘š β†’ ((π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ↔ (π‘”β€˜(π‘š + 1)) ∈ (π‘šπΉ(π‘”β€˜π‘š))))
2625rspccva 3581 . . . . . . . . . . . . . 14 ((βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ∧ π‘š ∈ β„•) β†’ (π‘”β€˜(π‘š + 1)) ∈ (π‘šπΉ(π‘”β€˜π‘š)))
2713, 26sylan 581 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘”β€˜(π‘š + 1)) ∈ (π‘šπΉ(π‘”β€˜π‘š)))
286ffvelcdmda 7036 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘”β€˜π‘š) ∈ (𝑋 Γ— ℝ+))
297, 1, 8bcthlem1 24691 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (π‘”β€˜π‘š) ∈ (𝑋 Γ— ℝ+))) β†’ ((π‘”β€˜(π‘š + 1)) ∈ (π‘šπΉ(π‘”β€˜π‘š)) ↔ ((π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘š)) βˆ– (π‘€β€˜π‘š)))))
3029expr 458 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((π‘”β€˜π‘š) ∈ (𝑋 Γ— ℝ+) β†’ ((π‘”β€˜(π‘š + 1)) ∈ (π‘šπΉ(π‘”β€˜π‘š)) ↔ ((π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘š)) βˆ– (π‘€β€˜π‘š))))))
3128, 30mpd 15 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((π‘”β€˜(π‘š + 1)) ∈ (π‘šπΉ(π‘”β€˜π‘š)) ↔ ((π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘š)) βˆ– (π‘€β€˜π‘š)))))
3227, 31mpbid 231 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘š)) βˆ– (π‘€β€˜π‘š))))
3332simp2d 1144 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š))
3433adantlr 714 . . . . . . . . . 10 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š))
3532simp1d 1143 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+))
36 xp2nd 7955 . . . . . . . . . . . . . 14 ((π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ+)
3735, 36syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ+)
3837rpred 12958 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ)
3938adantlr 714 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ)
40 nnrecre 12196 . . . . . . . . . . . 12 (π‘š ∈ β„• β†’ (1 / π‘š) ∈ ℝ)
4140adantl 483 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (1 / π‘š) ∈ ℝ)
42 rpre 12924 . . . . . . . . . . . 12 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ)
4342ad2antlr 726 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ π‘Ÿ ∈ ℝ)
44 lttr 11232 . . . . . . . . . . 11 (((2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ ∧ (1 / π‘š) ∈ ℝ ∧ π‘Ÿ ∈ ℝ) β†’ (((2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š) ∧ (1 / π‘š) < π‘Ÿ) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) < π‘Ÿ))
4539, 41, 43, 44syl3anc 1372 . . . . . . . . . 10 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (((2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š) ∧ (1 / π‘š) < π‘Ÿ) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) < π‘Ÿ))
4634, 45mpand 694 . . . . . . . . 9 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ ((1 / π‘š) < π‘Ÿ β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) < π‘Ÿ))
47 2fveq3 6848 . . . . . . . . . . 11 (𝑛 = (π‘š + 1) β†’ (2nd β€˜(π‘”β€˜π‘›)) = (2nd β€˜(π‘”β€˜(π‘š + 1))))
4847breq1d 5116 . . . . . . . . . 10 (𝑛 = (π‘š + 1) β†’ ((2nd β€˜(π‘”β€˜π‘›)) < π‘Ÿ ↔ (2nd β€˜(π‘”β€˜(π‘š + 1))) < π‘Ÿ))
4948rspcev 3582 . . . . . . . . 9 (((π‘š + 1) ∈ β„• ∧ (2nd β€˜(π‘”β€˜(π‘š + 1))) < π‘Ÿ) β†’ βˆƒπ‘› ∈ β„• (2nd β€˜(π‘”β€˜π‘›)) < π‘Ÿ)
5020, 46, 49syl6an 683 . . . . . . . 8 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ ((1 / π‘š) < π‘Ÿ β†’ βˆƒπ‘› ∈ β„• (2nd β€˜(π‘”β€˜π‘›)) < π‘Ÿ))
5150rexlimdva 3153 . . . . . . 7 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ (βˆƒπ‘š ∈ β„• (1 / π‘š) < π‘Ÿ β†’ βˆƒπ‘› ∈ β„• (2nd β€˜(π‘”β€˜π‘›)) < π‘Ÿ))
5218, 51mpd 15 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ βˆƒπ‘› ∈ β„• (2nd β€˜(π‘”β€˜π‘›)) < π‘Ÿ)
5352ralrimiva 3144 . . . . 5 (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘› ∈ β„• (2nd β€˜(π‘”β€˜π‘›)) < π‘Ÿ)
545, 6, 14, 53caubl 24675 . . . 4 (πœ‘ β†’ (1st ∘ 𝑔) ∈ (Cauβ€˜π·))
557cmetcau 24656 . . . 4 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (1st ∘ 𝑔) ∈ (Cauβ€˜π·)) β†’ (1st ∘ 𝑔) ∈ dom (β‡π‘‘β€˜π½))
561, 54, 55syl2anc 585 . . 3 (πœ‘ β†’ (1st ∘ 𝑔) ∈ dom (β‡π‘‘β€˜π½))
57 fo1st 7942 . . . . . 6 1st :V–ontoβ†’V
58 fofun 6758 . . . . . 6 (1st :V–ontoβ†’V β†’ Fun 1st )
5957, 58ax-mp 5 . . . . 5 Fun 1st
60 vex 3450 . . . . 5 𝑔 ∈ V
61 cofunexg 7882 . . . . 5 ((Fun 1st ∧ 𝑔 ∈ V) β†’ (1st ∘ 𝑔) ∈ V)
6259, 60, 61mp2an 691 . . . 4 (1st ∘ 𝑔) ∈ V
6362eldm 5857 . . 3 ((1st ∘ 𝑔) ∈ dom (β‡π‘‘β€˜π½) ↔ βˆƒπ‘₯(1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯)
6456, 63sylib 217 . 2 (πœ‘ β†’ βˆƒπ‘₯(1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯)
65 1nn 12165 . . . . . 6 1 ∈ β„•
667, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 24693 . . . . . 6 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯ ∧ 1 ∈ β„•) β†’ π‘₯ ∈ ((ballβ€˜π·)β€˜(π‘”β€˜1)))
6765, 66mp3an3 1451 . . . . 5 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ π‘₯ ∈ ((ballβ€˜π·)β€˜(π‘”β€˜1)))
6812fveq2d 6847 . . . . . . 7 (πœ‘ β†’ ((ballβ€˜π·)β€˜(π‘”β€˜1)) = ((ballβ€˜π·)β€˜βŸ¨πΆ, π‘…βŸ©))
69 df-ov 7361 . . . . . . 7 (𝐢(ballβ€˜π·)𝑅) = ((ballβ€˜π·)β€˜βŸ¨πΆ, π‘…βŸ©)
7068, 69eqtr4di 2795 . . . . . 6 (πœ‘ β†’ ((ballβ€˜π·)β€˜(π‘”β€˜1)) = (𝐢(ballβ€˜π·)𝑅))
7170adantr 482 . . . . 5 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜1)) = (𝐢(ballβ€˜π·)𝑅))
7267, 71eleqtrd 2840 . . . 4 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ π‘₯ ∈ (𝐢(ballβ€˜π·)𝑅))
737mopntop 23796 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
745, 73syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐽 ∈ Top)
7574adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐽 ∈ Top)
765adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
77 xp1st 7954 . . . . . . . . . . . . . . 15 ((π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(π‘”β€˜(π‘š + 1))) ∈ 𝑋)
7835, 77syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (1st β€˜(π‘”β€˜(π‘š + 1))) ∈ 𝑋)
7937rpxrd 12959 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ*)
80 blssm 23774 . . . . . . . . . . . . . 14 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜(π‘”β€˜(π‘š + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ*) β†’ ((1st β€˜(π‘”β€˜(π‘š + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(π‘š + 1)))) βŠ† 𝑋)
8176, 78, 79, 80syl3anc 1372 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((1st β€˜(π‘”β€˜(π‘š + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(π‘š + 1)))) βŠ† 𝑋)
82 df-ov 7361 . . . . . . . . . . . . . 14 ((1st β€˜(π‘”β€˜(π‘š + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(π‘š + 1)))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(π‘š + 1))), (2nd β€˜(π‘”β€˜(π‘š + 1)))⟩)
83 1st2nd2 7961 . . . . . . . . . . . . . . . 16 ((π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (π‘”β€˜(π‘š + 1)) = ⟨(1st β€˜(π‘”β€˜(π‘š + 1))), (2nd β€˜(π‘”β€˜(π‘š + 1)))⟩)
8435, 83syl 17 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘”β€˜(π‘š + 1)) = ⟨(1st β€˜(π‘”β€˜(π‘š + 1))), (2nd β€˜(π‘”β€˜(π‘š + 1)))⟩)
8584fveq2d 6847 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(π‘š + 1))), (2nd β€˜(π‘”β€˜(π‘š + 1)))⟩))
8682, 85eqtr4id 2796 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((1st β€˜(π‘”β€˜(π‘š + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(π‘š + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))))
877mopnuni 23797 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
885, 87syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
8988adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝑋 = βˆͺ 𝐽)
9081, 86, 893sstr3d 3991 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))) βŠ† βˆͺ 𝐽)
91 eqid 2737 . . . . . . . . . . . . 13 βˆͺ 𝐽 = βˆͺ 𝐽
9291sscls 22410 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))) βŠ† βˆͺ 𝐽) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1)))))
9375, 90, 92syl2anc 585 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1)))))
9432simp3d 1145 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘š)) βˆ– (π‘€β€˜π‘š)))
9593, 94sstrd 3955 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘š)) βˆ– (π‘€β€˜π‘š)))
96953adant2 1132 . . . . . . . . 9 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯ ∧ π‘š ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘š)) βˆ– (π‘€β€˜π‘š)))
977, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 24693 . . . . . . . . . 10 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯ ∧ (π‘š + 1) ∈ β„•) β†’ π‘₯ ∈ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))))
9819, 97syl3an3 1166 . . . . . . . . 9 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯ ∧ π‘š ∈ β„•) β†’ π‘₯ ∈ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))))
9996, 98sseldd 3946 . . . . . . . 8 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯ ∧ π‘š ∈ β„•) β†’ π‘₯ ∈ (((ballβ€˜π·)β€˜(π‘”β€˜π‘š)) βˆ– (π‘€β€˜π‘š)))
10099eldifbd 3924 . . . . . . 7 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯ ∧ π‘š ∈ β„•) β†’ Β¬ π‘₯ ∈ (π‘€β€˜π‘š))
1011003expa 1119 . . . . . 6 (((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) ∧ π‘š ∈ β„•) β†’ Β¬ π‘₯ ∈ (π‘€β€˜π‘š))
102101ralrimiva 3144 . . . . 5 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ βˆ€π‘š ∈ β„• Β¬ π‘₯ ∈ (π‘€β€˜π‘š))
103 eluni2 4870 . . . . . . . . 9 (π‘₯ ∈ βˆͺ ran 𝑀 ↔ βˆƒπ‘¦ ∈ ran 𝑀 π‘₯ ∈ 𝑦)
1049ffnd 6670 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 Fn β„•)
105 eleq2 2827 . . . . . . . . . . 11 (𝑦 = (π‘€β€˜π‘š) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ (π‘€β€˜π‘š)))
106105rexrn 7038 . . . . . . . . . 10 (𝑀 Fn β„• β†’ (βˆƒπ‘¦ ∈ ran 𝑀 π‘₯ ∈ 𝑦 ↔ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘€β€˜π‘š)))
107104, 106syl 17 . . . . . . . . 9 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ran 𝑀 π‘₯ ∈ 𝑦 ↔ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘€β€˜π‘š)))
108103, 107bitrid 283 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ βˆͺ ran 𝑀 ↔ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘€β€˜π‘š)))
109108notbid 318 . . . . . . 7 (πœ‘ β†’ (Β¬ π‘₯ ∈ βˆͺ ran 𝑀 ↔ Β¬ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘€β€˜π‘š)))
110 ralnex 3076 . . . . . . 7 (βˆ€π‘š ∈ β„• Β¬ π‘₯ ∈ (π‘€β€˜π‘š) ↔ Β¬ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘€β€˜π‘š))
111109, 110bitr4di 289 . . . . . 6 (πœ‘ β†’ (Β¬ π‘₯ ∈ βˆͺ ran 𝑀 ↔ βˆ€π‘š ∈ β„• Β¬ π‘₯ ∈ (π‘€β€˜π‘š)))
112111biimpar 479 . . . . 5 ((πœ‘ ∧ βˆ€π‘š ∈ β„• Β¬ π‘₯ ∈ (π‘€β€˜π‘š)) β†’ Β¬ π‘₯ ∈ βˆͺ ran 𝑀)
113102, 112syldan 592 . . . 4 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ Β¬ π‘₯ ∈ βˆͺ ran 𝑀)
11472, 113eldifd 3922 . . 3 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ π‘₯ ∈ ((𝐢(ballβ€˜π·)𝑅) βˆ– βˆͺ ran 𝑀))
115114ne0d 4296 . 2 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ ((𝐢(ballβ€˜π·)𝑅) βˆ– βˆͺ ran 𝑀) β‰  βˆ…)
11664, 115exlimddv 1939 1 (πœ‘ β†’ ((𝐢(ballβ€˜π·)𝑅) βˆ– βˆͺ ran 𝑀) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3446   βˆ– cdif 3908   βŠ† wss 3911  βˆ…c0 4283  βŸ¨cop 4593  βˆͺ cuni 4866   class class class wbr 5106  {copab 5168   Γ— cxp 5632  dom cdm 5634  ran crn 5635   ∘ ccom 5638  Fun wfun 6491   Fn wfn 6492  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  β„cr 11051  0cc0 11052  1c1 11053   + caddc 11055  β„*cxr 11189   < clt 11190   / cdiv 11813  β„•cn 12154  β„+crp 12916  βˆžMetcxmet 20784  Metcmet 20785  ballcbl 20786  MetOpencmopn 20789  Topctop 22245  Clsdccld 22370  clsccl 22372  β‡π‘‘clm 22580  Cauccau 24620  CMetccmet 24621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8649  df-map 8768  df-pm 8769  df-en 8885  df-dom 8886  df-sdom 8887  df-sup 9379  df-inf 9380  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-n0 12415  df-z 12501  df-uz 12765  df-q 12875  df-rp 12917  df-xneg 13034  df-xadd 13035  df-xmul 13036  df-ico 13271  df-rest 17305  df-topgen 17326  df-psmet 20791  df-xmet 20792  df-met 20793  df-bl 20794  df-mopn 20795  df-fbas 20796  df-fg 20797  df-top 22246  df-topon 22263  df-bases 22299  df-cld 22373  df-ntr 22374  df-cls 22375  df-nei 22452  df-lm 22583  df-fil 23200  df-fm 23292  df-flim 23293  df-flf 23294  df-cfil 24622  df-cau 24623  df-cmet 24624
This theorem is referenced by:  bcthlem5  24695
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