Step | Hyp | Ref
| Expression |
1 | | remet.1 |
. . . 4
β’ π· = ((abs β β )
βΎ (β Γ β)) |
2 | 1 | rexmet 14011 |
. . 3
β’ π· β
(βMetββ) |
3 | | tgioo.2 |
. . . 4
β’ π½ = (MetOpenβπ·) |
4 | 3 | mopnval 13912 |
. . 3
β’ (π· β
(βMetββ) β π½ = (topGenβran (ballβπ·))) |
5 | 2, 4 | ax-mp 5 |
. 2
β’ π½ = (topGenβran
(ballβπ·)) |
6 | | blex 13857 |
. . . . 5
β’ (π· β
(βMetββ) β (ballβπ·) β V) |
7 | 2, 6 | ax-mp 5 |
. . . 4
β’
(ballβπ·)
β V |
8 | 7 | rnex 4894 |
. . 3
β’ ran
(ballβπ·) β
V |
9 | 1 | blssioo 14015 |
. . 3
β’ ran
(ballβπ·) β ran
(,) |
10 | | elssuni 3837 |
. . . . . . 7
β’ (π£ β ran (,) β π£ β βͺ ran (,)) |
11 | | unirnioo 9972 |
. . . . . . 7
β’ β =
βͺ ran (,) |
12 | 10, 11 | sseqtrrdi 3204 |
. . . . . 6
β’ (π£ β ran (,) β π£ β
β) |
13 | | retopbas 13993 |
. . . . . . . . . 10
β’ ran (,)
β TopBases |
14 | 13 | a1i 9 |
. . . . . . . . 9
β’ ((π£ β ran (,) β§ π₯ β π£) β ran (,) β
TopBases) |
15 | | simpl 109 |
. . . . . . . . 9
β’ ((π£ β ran (,) β§ π₯ β π£) β π£ β ran (,)) |
16 | 12 | sselda 3155 |
. . . . . . . . . 10
β’ ((π£ β ran (,) β§ π₯ β π£) β π₯ β β) |
17 | | 1re 7955 |
. . . . . . . . . . . 12
β’ 1 β
β |
18 | 1 | bl2ioo 14012 |
. . . . . . . . . . . 12
β’ ((π₯ β β β§ 1 β
β) β (π₯(ballβπ·)1) = ((π₯ β 1)(,)(π₯ + 1))) |
19 | 17, 18 | mpan2 425 |
. . . . . . . . . . 11
β’ (π₯ β β β (π₯(ballβπ·)1) = ((π₯ β 1)(,)(π₯ + 1))) |
20 | | peano2rem 8223 |
. . . . . . . . . . . . 13
β’ (π₯ β β β (π₯ β 1) β
β) |
21 | 20 | rexrd 8006 |
. . . . . . . . . . . 12
β’ (π₯ β β β (π₯ β 1) β
β*) |
22 | | peano2re 8092 |
. . . . . . . . . . . . 13
β’ (π₯ β β β (π₯ + 1) β
β) |
23 | 22 | rexrd 8006 |
. . . . . . . . . . . 12
β’ (π₯ β β β (π₯ + 1) β
β*) |
24 | | ioorebasg 9974 |
. . . . . . . . . . . 12
β’ (((π₯ β 1) β
β* β§ (π₯ + 1) β β*) β
((π₯ β 1)(,)(π₯ + 1)) β ran
(,)) |
25 | 21, 23, 24 | syl2anc 411 |
. . . . . . . . . . 11
β’ (π₯ β β β ((π₯ β 1)(,)(π₯ + 1)) β ran
(,)) |
26 | 19, 25 | eqeltrd 2254 |
. . . . . . . . . 10
β’ (π₯ β β β (π₯(ballβπ·)1) β ran (,)) |
27 | 16, 26 | syl 14 |
. . . . . . . . 9
β’ ((π£ β ran (,) β§ π₯ β π£) β (π₯(ballβπ·)1) β ran (,)) |
28 | | simpr 110 |
. . . . . . . . . 10
β’ ((π£ β ran (,) β§ π₯ β π£) β π₯ β π£) |
29 | | 1rp 9656 |
. . . . . . . . . . . 12
β’ 1 β
β+ |
30 | | blcntr 13886 |
. . . . . . . . . . . 12
β’ ((π· β
(βMetββ) β§ π₯ β β β§ 1 β
β+) β π₯ β (π₯(ballβπ·)1)) |
31 | 2, 29, 30 | mp3an13 1328 |
. . . . . . . . . . 11
β’ (π₯ β β β π₯ β (π₯(ballβπ·)1)) |
32 | 16, 31 | syl 14 |
. . . . . . . . . 10
β’ ((π£ β ran (,) β§ π₯ β π£) β π₯ β (π₯(ballβπ·)1)) |
33 | 28, 32 | elind 3320 |
. . . . . . . . 9
β’ ((π£ β ran (,) β§ π₯ β π£) β π₯ β (π£ β© (π₯(ballβπ·)1))) |
34 | | basis2 13518 |
. . . . . . . . 9
β’ (((ran
(,) β TopBases β§ π£
β ran (,)) β§ ((π₯(ballβπ·)1) β ran (,) β§ π₯ β (π£ β© (π₯(ballβπ·)1)))) β βπ§ β ran (,)(π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1)))) |
35 | 14, 15, 27, 33, 34 | syl22anc 1239 |
. . . . . . . 8
β’ ((π£ β ran (,) β§ π₯ β π£) β βπ§ β ran (,)(π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1)))) |
36 | | ioof 9970 |
. . . . . . . . . . 11
β’
(,):(β* Γ β*)βΆπ«
β |
37 | | ffn 5365 |
. . . . . . . . . . 11
β’
((,):(β* Γ β*)βΆπ«
β β (,) Fn (β* Γ
β*)) |
38 | | ovelrn 6022 |
. . . . . . . . . . 11
β’ ((,) Fn
(β* Γ β*) β (π§ β ran (,) β βπ β β*
βπ β
β* π§ =
(π(,)π))) |
39 | 36, 37, 38 | mp2b 8 |
. . . . . . . . . 10
β’ (π§ β ran (,) β
βπ β
β* βπ β β* π§ = (π(,)π)) |
40 | | eleq2 2241 |
. . . . . . . . . . . . . . 15
β’ (π§ = (π(,)π) β (π₯ β π§ β π₯ β (π(,)π))) |
41 | | sseq1 3178 |
. . . . . . . . . . . . . . 15
β’ (π§ = (π(,)π) β (π§ β (π£ β© (π₯(ballβπ·)1)) β (π(,)π) β (π£ β© (π₯(ballβπ·)1)))) |
42 | 40, 41 | anbi12d 473 |
. . . . . . . . . . . . . 14
β’ (π§ = (π(,)π) β ((π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1))) β (π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))))) |
43 | | inss2 3356 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π£ β© (π₯(ballβπ·)1)) β (π₯(ballβπ·)1) |
44 | | sstr 3163 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π(,)π) β (π£ β© (π₯(ballβπ·)1)) β§ (π£ β© (π₯(ballβπ·)1)) β (π₯(ballβπ·)1)) β (π(,)π) β (π₯(ballβπ·)1)) |
45 | 43, 44 | mpan2 425 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π(,)π) β (π£ β© (π₯(ballβπ·)1)) β (π(,)π) β (π₯(ballβπ·)1)) |
46 | 45 | adantl 277 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π(,)π) β (π₯(ballβπ·)1)) |
47 | | elioore 9911 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π₯ β (π(,)π) β π₯ β β) |
48 | 47 | adantr 276 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β π₯ β β) |
49 | 48, 19 | syl 14 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π₯(ballβπ·)1) = ((π₯ β 1)(,)(π₯ + 1))) |
50 | 46, 49 | sseqtrd 3193 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π(,)π) β ((π₯ β 1)(,)(π₯ + 1))) |
51 | | dfss 3143 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π(,)π) β ((π₯ β 1)(,)(π₯ + 1)) β (π(,)π) = ((π(,)π) β© ((π₯ β 1)(,)(π₯ + 1)))) |
52 | 50, 51 | sylib 122 |
. . . . . . . . . . . . . . . . . 18
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π(,)π) = ((π(,)π) β© ((π₯ β 1)(,)(π₯ + 1)))) |
53 | | eliooxr 9926 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π₯ β (π(,)π) β (π β β* β§ π β
β*)) |
54 | 21, 23 | jca 306 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π₯ β β β ((π₯ β 1) β
β* β§ (π₯ + 1) β
β*)) |
55 | 47, 54 | syl 14 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π₯ β (π(,)π) β ((π₯ β 1) β β* β§
(π₯ + 1) β
β*)) |
56 | | iooinsup 11284 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π β β*
β§ π β
β*) β§ ((π₯ β 1) β β* β§
(π₯ + 1) β
β*)) β ((π(,)π) β© ((π₯ β 1)(,)(π₯ + 1))) = (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< ))) |
57 | 53, 55, 56 | syl2anc 411 |
. . . . . . . . . . . . . . . . . . 19
β’ (π₯ β (π(,)π) β ((π(,)π) β© ((π₯ β 1)(,)(π₯ + 1))) = (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< ))) |
58 | 57 | adantr 276 |
. . . . . . . . . . . . . . . . . 18
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β ((π(,)π) β© ((π₯ β 1)(,)(π₯ + 1))) = (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< ))) |
59 | 52, 58 | eqtrd 2210 |
. . . . . . . . . . . . . . . . 17
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π(,)π) = (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< ))) |
60 | | mnfxr 8013 |
. . . . . . . . . . . . . . . . . . . 20
β’ -β
β β* |
61 | 60 | a1i 9 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β -β β
β*) |
62 | 53 | adantr 276 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π β β* β§ π β
β*)) |
63 | 62 | simpld 112 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β π β β*) |
64 | 48, 21 | syl 14 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π₯ β 1) β
β*) |
65 | | xrmaxcl 11259 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β β*
β§ (π₯ β 1) β
β*) β sup({π, (π₯ β 1)}, β*, < )
β β*) |
66 | 63, 64, 65 | syl2anc 411 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β sup({π, (π₯ β 1)}, β*, < )
β β*) |
67 | 62 | simprd 114 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β π β β*) |
68 | 48, 22 | syl 14 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π₯ + 1) β β) |
69 | 68 | rexrd 8006 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π₯ + 1) β
β*) |
70 | | xrmincl 11273 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β β*
β§ (π₯ + 1) β
β*) β inf({π, (π₯ + 1)}, β*, < ) β
β*) |
71 | 67, 69, 70 | syl2anc 411 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β inf({π, (π₯ + 1)}, β*, < ) β
β*) |
72 | 47, 20 | syl 14 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π₯ β (π(,)π) β (π₯ β 1) β β) |
73 | 72 | adantr 276 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π₯ β 1) β β) |
74 | | mnflt 9782 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯ β 1) β β
β -β < (π₯
β 1)) |
75 | 73, 74 | syl 14 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β -β < (π₯ β 1)) |
76 | | xrmax2sup 11261 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π β β*
β§ (π₯ β 1) β
β*) β (π₯ β 1) β€ sup({π, (π₯ β 1)}, β*, <
)) |
77 | 63, 64, 76 | syl2anc 411 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π₯ β 1) β€ sup({π, (π₯ β 1)}, β*, <
)) |
78 | 61, 64, 66, 75, 77 | xrltletrd 9810 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β -β < sup({π, (π₯ β 1)}, β*, <
)) |
79 | | simpl 109 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β π₯ β (π(,)π)) |
80 | 79, 59 | eleqtrd 2256 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β π₯ β (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< ))) |
81 | | eliooxr 9926 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π₯ β (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< )) β (sup({π,
(π₯ β 1)},
β*, < ) β β* β§ inf({π, (π₯ + 1)}, β*, < ) β
β*)) |
82 | | elex2 2753 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π₯ β (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< )) β βπ€
π€ β (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< ))) |
83 | | ioom 10260 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
((sup({π, (π₯ β 1)},
β*, < ) β β* β§ inf({π, (π₯ + 1)}, β*, < ) β
β*) β (βπ€ π€ β (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< )) β sup({π,
(π₯ β 1)},
β*, < ) < inf({π, (π₯ + 1)}, β*, <
))) |
84 | 82, 83 | imbitrid 154 |
. . . . . . . . . . . . . . . . . . . . 21
β’
((sup({π, (π₯ β 1)},
β*, < ) β β* β§ inf({π, (π₯ + 1)}, β*, < ) β
β*) β (π₯ β (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< )) β sup({π,
(π₯ β 1)},
β*, < ) < inf({π, (π₯ + 1)}, β*, <
))) |
85 | 81, 84 | mpcom 36 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π₯ β (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< )) β sup({π,
(π₯ β 1)},
β*, < ) < inf({π, (π₯ + 1)}, β*, <
)) |
86 | 80, 85 | syl 14 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β sup({π, (π₯ β 1)}, β*, < )
< inf({π, (π₯ + 1)}, β*,
< )) |
87 | | xrre2 9820 |
. . . . . . . . . . . . . . . . . . 19
β’
(((-β β β* β§ sup({π, (π₯ β 1)}, β*, < )
β β* β§ inf({π, (π₯ + 1)}, β*, < ) β
β*) β§ (-β < sup({π, (π₯ β 1)}, β*, < )
β§ sup({π, (π₯ β 1)},
β*, < ) < inf({π, (π₯ + 1)}, β*, < ))) β
sup({π, (π₯ β 1)}, β*, < )
β β) |
88 | 61, 66, 71, 78, 86, 87 | syl32anc 1246 |
. . . . . . . . . . . . . . . . . 18
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β sup({π, (π₯ β 1)}, β*, < )
β β) |
89 | | mnfle 9791 |
. . . . . . . . . . . . . . . . . . . . 21
β’
(sup({π, (π₯ β 1)},
β*, < ) β β* β -β β€
sup({π, (π₯ β 1)}, β*, <
)) |
90 | 66, 89 | syl 14 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β -β β€ sup({π, (π₯ β 1)}, β*, <
)) |
91 | 61, 66, 71, 90, 86 | xrlelttrd 9809 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β -β < inf({π, (π₯ + 1)}, β*, <
)) |
92 | | xrmin2inf 11275 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β β*
β§ (π₯ + 1) β
β*) β inf({π, (π₯ + 1)}, β*, < ) β€
(π₯ + 1)) |
93 | 67, 69, 92 | syl2anc 411 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β inf({π, (π₯ + 1)}, β*, < ) β€
(π₯ + 1)) |
94 | | xrre 9819 |
. . . . . . . . . . . . . . . . . . 19
β’
(((inf({π, (π₯ + 1)}, β*,
< ) β β* β§ (π₯ + 1) β β) β§ (-β <
inf({π, (π₯ + 1)}, β*, < ) β§
inf({π, (π₯ + 1)}, β*, < ) β€
(π₯ + 1))) β inf({π, (π₯ + 1)}, β*, < ) β
β) |
95 | 71, 68, 91, 93, 94 | syl22anc 1239 |
. . . . . . . . . . . . . . . . . 18
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β inf({π, (π₯ + 1)}, β*, < ) β
β) |
96 | 1 | ioo2blex 14014 |
. . . . . . . . . . . . . . . . . 18
β’
((sup({π, (π₯ β 1)},
β*, < ) β β β§ inf({π, (π₯ + 1)}, β*, < ) β
β) β (sup({π,
(π₯ β 1)},
β*, < )(,)inf({π, (π₯ + 1)}, β*, < )) β
ran (ballβπ·)) |
97 | 88, 95, 96 | syl2anc 411 |
. . . . . . . . . . . . . . . . 17
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (sup({π, (π₯ β 1)}, β*, <
)(,)inf({π, (π₯ + 1)}, β*,
< )) β ran (ballβπ·)) |
98 | 59, 97 | eqeltrd 2254 |
. . . . . . . . . . . . . . . 16
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π(,)π) β ran (ballβπ·)) |
99 | | inss1 3355 |
. . . . . . . . . . . . . . . . . 18
β’ (π£ β© (π₯(ballβπ·)1)) β π£ |
100 | | sstr 3163 |
. . . . . . . . . . . . . . . . . 18
β’ (((π(,)π) β (π£ β© (π₯(ballβπ·)1)) β§ (π£ β© (π₯(ballβπ·)1)) β π£) β (π(,)π) β π£) |
101 | 99, 100 | mpan2 425 |
. . . . . . . . . . . . . . . . 17
β’ ((π(,)π) β (π£ β© (π₯(ballβπ·)1)) β (π(,)π) β π£) |
102 | 101 | adantl 277 |
. . . . . . . . . . . . . . . 16
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (π(,)π) β π£) |
103 | | sseq1 3178 |
. . . . . . . . . . . . . . . . . 18
β’ (π§ = (π(,)π) β (π§ β π£ β (π(,)π) β π£)) |
104 | 40, 103 | anbi12d 473 |
. . . . . . . . . . . . . . . . 17
β’ (π§ = (π(,)π) β ((π₯ β π§ β§ π§ β π£) β (π₯ β (π(,)π) β§ (π(,)π) β π£))) |
105 | 104 | rspcev 2841 |
. . . . . . . . . . . . . . . 16
β’ (((π(,)π) β ran (ballβπ·) β§ (π₯ β (π(,)π) β§ (π(,)π) β π£)) β βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π£)) |
106 | 98, 79, 102, 105 | syl12anc 1236 |
. . . . . . . . . . . . . . 15
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π£)) |
107 | | blssex 13900 |
. . . . . . . . . . . . . . . 16
β’ ((π· β
(βMetββ) β§ π₯ β β) β (βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π£) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£)) |
108 | 2, 48, 107 | sylancr 414 |
. . . . . . . . . . . . . . 15
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β (βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π£) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£)) |
109 | 106, 108 | mpbid 147 |
. . . . . . . . . . . . . 14
β’ ((π₯ β (π(,)π) β§ (π(,)π) β (π£ β© (π₯(ballβπ·)1))) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£) |
110 | 42, 109 | syl6bi 163 |
. . . . . . . . . . . . 13
β’ (π§ = (π(,)π) β ((π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1))) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£)) |
111 | 110 | a1i 9 |
. . . . . . . . . . . 12
β’ ((π β β*
β§ π β
β*) β (π§ = (π(,)π) β ((π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1))) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£))) |
112 | 111 | rexlimivv 2600 |
. . . . . . . . . . 11
β’
(βπ β
β* βπ β β* π§ = (π(,)π) β ((π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1))) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£)) |
113 | 112 | imp 124 |
. . . . . . . . . 10
β’
((βπ β
β* βπ β β* π§ = (π(,)π) β§ (π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1)))) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£) |
114 | 39, 113 | sylanb 284 |
. . . . . . . . 9
β’ ((π§ β ran (,) β§ (π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1)))) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£) |
115 | 114 | rexlimiva 2589 |
. . . . . . . 8
β’
(βπ§ β ran
(,)(π₯ β π§ β§ π§ β (π£ β© (π₯(ballβπ·)1))) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£) |
116 | 35, 115 | syl 14 |
. . . . . . 7
β’ ((π£ β ran (,) β§ π₯ β π£) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π£) |
117 | 116 | ralrimiva 2550 |
. . . . . 6
β’ (π£ β ran (,) β
βπ₯ β π£ βπ¦ β β+ (π₯(ballβπ·)π¦) β π£) |
118 | 3 | elmopn2 13919 |
. . . . . . 7
β’ (π· β
(βMetββ) β (π£ β π½ β (π£ β β β§ βπ₯ β π£ βπ¦ β β+ (π₯(ballβπ·)π¦) β π£))) |
119 | 2, 118 | ax-mp 5 |
. . . . . 6
β’ (π£ β π½ β (π£ β β β§ βπ₯ β π£ βπ¦ β β+ (π₯(ballβπ·)π¦) β π£)) |
120 | 12, 117, 119 | sylanbrc 417 |
. . . . 5
β’ (π£ β ran (,) β π£ β π½) |
121 | 120 | ssriv 3159 |
. . . 4
β’ ran (,)
β π½ |
122 | 121, 5 | sseqtri 3189 |
. . 3
β’ ran (,)
β (topGenβran (ballβπ·)) |
123 | | 2basgeng 13552 |
. . 3
β’ ((ran
(ballβπ·) β V
β§ ran (ballβπ·)
β ran (,) β§ ran (,) β (topGenβran (ballβπ·))) β (topGenβran
(ballβπ·)) =
(topGenβran (,))) |
124 | 8, 9, 122, 123 | mp3an 1337 |
. 2
β’
(topGenβran (ballβπ·)) = (topGenβran (,)) |
125 | 5, 124 | eqtr2i 2199 |
1
β’
(topGenβran (,)) = π½ |