Step | Hyp | Ref
| Expression |
1 | | metss2.2 |
. . . . . . 7
β’ (π β π· β (Metβπ)) |
2 | 1 | ad2antrr 725 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β π· β (Metβπ)) |
3 | | simplrl 776 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β π₯ β π) |
4 | | simpr 486 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β π¦ β π) |
5 | | metcl 23701 |
. . . . . 6
β’ ((π· β (Metβπ) β§ π₯ β π β§ π¦ β π) β (π₯π·π¦) β β) |
6 | 2, 3, 4, 5 | syl3anc 1372 |
. . . . 5
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β (π₯π·π¦) β β) |
7 | | simplrr 777 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β π β
β+) |
8 | 7 | rpred 12964 |
. . . . 5
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β π β β) |
9 | | metss2.3 |
. . . . . 6
β’ (π β π
β
β+) |
10 | 9 | ad2antrr 725 |
. . . . 5
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β π
β
β+) |
11 | 6, 8, 10 | ltmuldiv2d 13012 |
. . . 4
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β ((π
Β· (π₯π·π¦)) < π β (π₯π·π¦) < (π / π
))) |
12 | | metss2.4 |
. . . . . . 7
β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯πΆπ¦) β€ (π
Β· (π₯π·π¦))) |
13 | 12 | anassrs 469 |
. . . . . 6
β’ (((π β§ π₯ β π) β§ π¦ β π) β (π₯πΆπ¦) β€ (π
Β· (π₯π·π¦))) |
14 | 13 | adantlrr 720 |
. . . . 5
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β (π₯πΆπ¦) β€ (π
Β· (π₯π·π¦))) |
15 | | metss2.1 |
. . . . . . . 8
β’ (π β πΆ β (Metβπ)) |
16 | 15 | ad2antrr 725 |
. . . . . . 7
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β πΆ β (Metβπ)) |
17 | | metcl 23701 |
. . . . . . 7
β’ ((πΆ β (Metβπ) β§ π₯ β π β§ π¦ β π) β (π₯πΆπ¦) β β) |
18 | 16, 3, 4, 17 | syl3anc 1372 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β (π₯πΆπ¦) β β) |
19 | 10 | rpred 12964 |
. . . . . . 7
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β π
β β) |
20 | 19, 6 | remulcld 11192 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β (π
Β· (π₯π·π¦)) β β) |
21 | | lelttr 11252 |
. . . . . 6
β’ (((π₯πΆπ¦) β β β§ (π
Β· (π₯π·π¦)) β β β§ π β β) β (((π₯πΆπ¦) β€ (π
Β· (π₯π·π¦)) β§ (π
Β· (π₯π·π¦)) < π) β (π₯πΆπ¦) < π)) |
22 | 18, 20, 8, 21 | syl3anc 1372 |
. . . . 5
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β (((π₯πΆπ¦) β€ (π
Β· (π₯π·π¦)) β§ (π
Β· (π₯π·π¦)) < π) β (π₯πΆπ¦) < π)) |
23 | 14, 22 | mpand 694 |
. . . 4
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β ((π
Β· (π₯π·π¦)) < π β (π₯πΆπ¦) < π)) |
24 | 11, 23 | sylbird 260 |
. . 3
β’ (((π β§ (π₯ β π β§ π β β+)) β§ π¦ β π) β ((π₯π·π¦) < (π / π
) β (π₯πΆπ¦) < π)) |
25 | 24 | ss2rabdv 4038 |
. 2
β’ ((π β§ (π₯ β π β§ π β β+)) β {π¦ β π β£ (π₯π·π¦) < (π / π
)} β {π¦ β π β£ (π₯πΆπ¦) < π}) |
26 | | metxmet 23703 |
. . . . 5
β’ (π· β (Metβπ) β π· β (βMetβπ)) |
27 | 1, 26 | syl 17 |
. . . 4
β’ (π β π· β (βMetβπ)) |
28 | 27 | adantr 482 |
. . 3
β’ ((π β§ (π₯ β π β§ π β β+)) β π· β (βMetβπ)) |
29 | | simprl 770 |
. . 3
β’ ((π β§ (π₯ β π β§ π β β+)) β π₯ β π) |
30 | | simpr 486 |
. . . . 5
β’ ((π₯ β π β§ π β β+) β π β
β+) |
31 | | rpdivcl 12947 |
. . . . 5
β’ ((π β β+
β§ π
β
β+) β (π / π
) β
β+) |
32 | 30, 9, 31 | syl2anr 598 |
. . . 4
β’ ((π β§ (π₯ β π β§ π β β+)) β (π / π
) β
β+) |
33 | 32 | rpxrd 12965 |
. . 3
β’ ((π β§ (π₯ β π β§ π β β+)) β (π / π
) β
β*) |
34 | | blval 23755 |
. . 3
β’ ((π· β (βMetβπ) β§ π₯ β π β§ (π / π
) β β*) β (π₯(ballβπ·)(π / π
)) = {π¦ β π β£ (π₯π·π¦) < (π / π
)}) |
35 | 28, 29, 33, 34 | syl3anc 1372 |
. 2
β’ ((π β§ (π₯ β π β§ π β β+)) β (π₯(ballβπ·)(π / π
)) = {π¦ β π β£ (π₯π·π¦) < (π / π
)}) |
36 | | metxmet 23703 |
. . . . 5
β’ (πΆ β (Metβπ) β πΆ β (βMetβπ)) |
37 | 15, 36 | syl 17 |
. . . 4
β’ (π β πΆ β (βMetβπ)) |
38 | 37 | adantr 482 |
. . 3
β’ ((π β§ (π₯ β π β§ π β β+)) β πΆ β (βMetβπ)) |
39 | | rpxr 12931 |
. . . 4
β’ (π β β+
β π β
β*) |
40 | 39 | ad2antll 728 |
. . 3
β’ ((π β§ (π₯ β π β§ π β β+)) β π β
β*) |
41 | | blval 23755 |
. . 3
β’ ((πΆ β (βMetβπ) β§ π₯ β π β§ π β β*) β (π₯(ballβπΆ)π) = {π¦ β π β£ (π₯πΆπ¦) < π}) |
42 | 38, 29, 40, 41 | syl3anc 1372 |
. 2
β’ ((π β§ (π₯ β π β§ π β β+)) β (π₯(ballβπΆ)π) = {π¦ β π β£ (π₯πΆπ¦) < π}) |
43 | 25, 35, 42 | 3sstr4d 3996 |
1
β’ ((π β§ (π₯ β π β§ π β β+)) β (π₯(ballβπ·)(π / π
)) β (π₯(ballβπΆ)π)) |