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Theorem bcthlem4 24851
Description: Lemma for bcth 24853. 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 24810 . . . . . . 7 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
31, 2syl 17 . . . . . 6 (πœ‘ β†’ 𝐷 ∈ (Metβ€˜π‘‹))
4 metxmet 23847 . . . . . 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 24849 . . . . 5 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
15 elrp 12978 . . . . . . . . 9 (π‘Ÿ ∈ ℝ+ ↔ (π‘Ÿ ∈ ℝ ∧ 0 < π‘Ÿ))
16 nnrecl 12472 . . . . . . . . 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 482 . . . . . . 7 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ βˆƒπ‘š ∈ β„• (1 / π‘š) < π‘Ÿ)
19 peano2nn 12226 . . . . . . . . . 10 (π‘š ∈ β„• β†’ (π‘š + 1) ∈ β„•)
2019adantl 482 . . . . . . . . 9 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (π‘š + 1) ∈ β„•)
21 fvoveq1 7434 . . . . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . 17 (π‘˜ = π‘š β†’ (π‘”β€˜π‘˜) = (π‘”β€˜π‘š))
2422, 23oveq12d 7429 . . . . . . . . . . . . . . . 16 (π‘˜ = π‘š β†’ (π‘˜πΉ(π‘”β€˜π‘˜)) = (π‘šπΉ(π‘”β€˜π‘š)))
2521, 24eleq12d 2827 . . . . . . . . . . . . . . 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 3611 . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘”β€˜(π‘š + 1)) ∈ (π‘šπΉ(π‘”β€˜π‘š)))
286ffvelcdmda 7086 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘”β€˜π‘š) ∈ (𝑋 Γ— ℝ+))
297, 1, 8bcthlem1 24848 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 1143 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š))
3433adantlr 713 . . . . . . . . . 10 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) < (1 / π‘š))
3532simp1d 1142 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘”β€˜(π‘š + 1)) ∈ (𝑋 Γ— ℝ+))
36 xp2nd 8010 . . . . . . . . . . . . . 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 13018 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ)
3938adantlr 713 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ)
40 nnrecre 12256 . . . . . . . . . . . 12 (π‘š ∈ β„• β†’ (1 / π‘š) ∈ ℝ)
4140adantl 482 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (1 / π‘š) ∈ ℝ)
42 rpre 12984 . . . . . . . . . . . 12 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ)
4342ad2antlr 725 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ π‘Ÿ ∈ ℝ)
44 lttr 11292 . . . . . . . . . . 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 1371 . . . . . . . . . 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 693 . . . . . . . . 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 6896 . . . . . . . . . . 11 (𝑛 = (π‘š + 1) β†’ (2nd β€˜(π‘”β€˜π‘›)) = (2nd β€˜(π‘”β€˜(π‘š + 1))))
4847breq1d 5158 . . . . . . . . . 10 (𝑛 = (π‘š + 1) β†’ ((2nd β€˜(π‘”β€˜π‘›)) < π‘Ÿ ↔ (2nd β€˜(π‘”β€˜(π‘š + 1))) < π‘Ÿ))
4948rspcev 3612 . . . . . . . . 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 682 . . . . . . . 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 3155 . . . . . . 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 3146 . . . . 5 (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘› ∈ β„• (2nd β€˜(π‘”β€˜π‘›)) < π‘Ÿ)
545, 6, 14, 53caubl 24832 . . . 4 (πœ‘ β†’ (1st ∘ 𝑔) ∈ (Cauβ€˜π·))
557cmetcau 24813 . . . 4 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (1st ∘ 𝑔) ∈ (Cauβ€˜π·)) β†’ (1st ∘ 𝑔) ∈ dom (β‡π‘‘β€˜π½))
561, 54, 55syl2anc 584 . . 3 (πœ‘ β†’ (1st ∘ 𝑔) ∈ dom (β‡π‘‘β€˜π½))
57 fo1st 7997 . . . . . 6 1st :V–ontoβ†’V
58 fofun 6806 . . . . . 6 (1st :V–ontoβ†’V β†’ Fun 1st )
5957, 58ax-mp 5 . . . . 5 Fun 1st
60 vex 3478 . . . . 5 𝑔 ∈ V
61 cofunexg 7937 . . . . 5 ((Fun 1st ∧ 𝑔 ∈ V) β†’ (1st ∘ 𝑔) ∈ V)
6259, 60, 61mp2an 690 . . . 4 (1st ∘ 𝑔) ∈ V
6362eldm 5900 . . 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 12225 . . . . . 6 1 ∈ β„•
667, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 24850 . . . . . 6 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯ ∧ 1 ∈ β„•) β†’ π‘₯ ∈ ((ballβ€˜π·)β€˜(π‘”β€˜1)))
6765, 66mp3an3 1450 . . . . 5 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ π‘₯ ∈ ((ballβ€˜π·)β€˜(π‘”β€˜1)))
6812fveq2d 6895 . . . . . . 7 (πœ‘ β†’ ((ballβ€˜π·)β€˜(π‘”β€˜1)) = ((ballβ€˜π·)β€˜βŸ¨πΆ, π‘…βŸ©))
69 df-ov 7414 . . . . . . 7 (𝐢(ballβ€˜π·)𝑅) = ((ballβ€˜π·)β€˜βŸ¨πΆ, π‘…βŸ©)
7068, 69eqtr4di 2790 . . . . . 6 (πœ‘ β†’ ((ballβ€˜π·)β€˜(π‘”β€˜1)) = (𝐢(ballβ€˜π·)𝑅))
7170adantr 481 . . . . 5 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜1)) = (𝐢(ballβ€˜π·)𝑅))
7267, 71eleqtrd 2835 . . . 4 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ π‘₯ ∈ (𝐢(ballβ€˜π·)𝑅))
737mopntop 23953 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
745, 73syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐽 ∈ Top)
7574adantr 481 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐽 ∈ Top)
765adantr 481 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
77 xp1st 8009 . . . . . . . . . . . . . . 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 13019 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (2nd β€˜(π‘”β€˜(π‘š + 1))) ∈ ℝ*)
80 blssm 23931 . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . 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 7414 . . . . . . . . . . . . . 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 8016 . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . 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 23954 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
885, 87syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
8988adantr 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝑋 = βˆͺ 𝐽)
9081, 86, 893sstr3d 4028 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(π‘š + 1))) βŠ† βˆͺ 𝐽)
91 eqid 2732 . . . . . . . . . . . . 13 βˆͺ 𝐽 = βˆͺ 𝐽
9291sscls 22567 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 1144 . . . . . . . . . . 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 3992 . . . . . . . . . 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 1131 . . . . . . . . 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 24850 . . . . . . . . . 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 1165 . . . . . . . . 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 3983 . . . . . . . 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 3961 . . . . . . 7 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯ ∧ π‘š ∈ β„•) β†’ Β¬ π‘₯ ∈ (π‘€β€˜π‘š))
1011003expa 1118 . . . . . 6 (((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) ∧ π‘š ∈ β„•) β†’ Β¬ π‘₯ ∈ (π‘€β€˜π‘š))
102101ralrimiva 3146 . . . . 5 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ βˆ€π‘š ∈ β„• Β¬ π‘₯ ∈ (π‘€β€˜π‘š))
103 eluni2 4912 . . . . . . . . 9 (π‘₯ ∈ βˆͺ ran 𝑀 ↔ βˆƒπ‘¦ ∈ ran 𝑀 π‘₯ ∈ 𝑦)
1049ffnd 6718 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 Fn β„•)
105 eleq2 2822 . . . . . . . . . . 11 (𝑦 = (π‘€β€˜π‘š) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ (π‘€β€˜π‘š)))
106105rexrn 7088 . . . . . . . . . 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 282 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ βˆͺ ran 𝑀 ↔ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘€β€˜π‘š)))
109108notbid 317 . . . . . . 7 (πœ‘ β†’ (Β¬ π‘₯ ∈ βˆͺ ran 𝑀 ↔ Β¬ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘€β€˜π‘š)))
110 ralnex 3072 . . . . . . 7 (βˆ€π‘š ∈ β„• Β¬ π‘₯ ∈ (π‘€β€˜π‘š) ↔ Β¬ βˆƒπ‘š ∈ β„• π‘₯ ∈ (π‘€β€˜π‘š))
111109, 110bitr4di 288 . . . . . 6 (πœ‘ β†’ (Β¬ π‘₯ ∈ βˆͺ ran 𝑀 ↔ βˆ€π‘š ∈ β„• Β¬ π‘₯ ∈ (π‘€β€˜π‘š)))
112111biimpar 478 . . . . 5 ((πœ‘ ∧ βˆ€π‘š ∈ β„• Β¬ π‘₯ ∈ (π‘€β€˜π‘š)) β†’ Β¬ π‘₯ ∈ βˆͺ ran 𝑀)
113102, 112syldan 591 . . . 4 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ Β¬ π‘₯ ∈ βˆͺ ran 𝑀)
11472, 113eldifd 3959 . . 3 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ π‘₯ ∈ ((𝐢(ballβ€˜π·)𝑅) βˆ– βˆͺ ran 𝑀))
115114ne0d 4335 . 2 ((πœ‘ ∧ (1st ∘ 𝑔)(β‡π‘‘β€˜π½)π‘₯) β†’ ((𝐢(ballβ€˜π·)𝑅) βˆ– βˆͺ ran 𝑀) β‰  βˆ…)
11664, 115exlimddv 1938 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 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆ– cdif 3945   βŠ† wss 3948  βˆ…c0 4322  βŸ¨cop 4634  βˆͺ cuni 4908   class class class wbr 5148  {copab 5210   Γ— cxp 5674  dom cdm 5676  ran crn 5677   ∘ ccom 5680  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115  β„*cxr 11249   < clt 11250   / cdiv 11873  β„•cn 12214  β„+crp 12976  βˆžMetcxmet 20935  Metcmet 20936  ballcbl 20937  MetOpencmopn 20940  Topctop 22402  Clsdccld 22527  clsccl 22529  β‡π‘‘clm 22737  Cauccau 24777  CMetccmet 24778
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-ico 13332  df-rest 17370  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-fbas 20947  df-fg 20948  df-top 22403  df-topon 22420  df-bases 22456  df-cld 22530  df-ntr 22531  df-cls 22532  df-nei 22609  df-lm 22740  df-fil 23357  df-fm 23449  df-flim 23450  df-flf 23451  df-cfil 24779  df-cau 24780  df-cmet 24781
This theorem is referenced by:  bcthlem5  24852
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