Step | Hyp | Ref
| Expression |
1 | | cfili 24655 |
. . 3
β’ ((πΉ β (CauFilβπ·) β§ π
β β+) β
βπ β πΉ βπ₯ β π βπ¦ β π (π₯π·π¦) < π
) |
2 | 1 | 3adant1 1131 |
. 2
β’ ((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β
βπ β πΉ βπ₯ β π βπ¦ β π (π₯π·π¦) < π
) |
3 | | cfilfil 24654 |
. . . . . . 7
β’ ((π· β (βMetβπ) β§ πΉ β (CauFilβπ·)) β πΉ β (Filβπ)) |
4 | 3 | 3adant3 1133 |
. . . . . 6
β’ ((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β πΉ β (Filβπ)) |
5 | | fileln0 23224 |
. . . . . 6
β’ ((πΉ β (Filβπ) β§ π β πΉ) β π β β
) |
6 | 4, 5 | sylan 581 |
. . . . 5
β’ (((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β π β β
) |
7 | | r19.2z 4456 |
. . . . . 6
β’ ((π β β
β§
βπ₯ β π βπ¦ β π (π₯π·π¦) < π
) β βπ₯ β π βπ¦ β π (π₯π·π¦) < π
) |
8 | 7 | ex 414 |
. . . . 5
β’ (π β β
β
(βπ₯ β π βπ¦ β π (π₯π·π¦) < π
β βπ₯ β π βπ¦ β π (π₯π·π¦) < π
)) |
9 | 6, 8 | syl 17 |
. . . 4
β’ (((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β (βπ₯ β π βπ¦ β π (π₯π·π¦) < π
β βπ₯ β π βπ¦ β π (π₯π·π¦) < π
)) |
10 | | filelss 23226 |
. . . . . 6
β’ ((πΉ β (Filβπ) β§ π β πΉ) β π β π) |
11 | 4, 10 | sylan 581 |
. . . . 5
β’ (((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β π β π) |
12 | | ssrexv 4015 |
. . . . 5
β’ (π β π β (βπ₯ β π βπ¦ β π (π₯π·π¦) < π
β βπ₯ β π βπ¦ β π (π₯π·π¦) < π
)) |
13 | 11, 12 | syl 17 |
. . . 4
β’ (((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β (βπ₯ β π βπ¦ β π (π₯π·π¦) < π
β βπ₯ β π βπ¦ β π (π₯π·π¦) < π
)) |
14 | | dfss3 3936 |
. . . . . . 7
β’ (π β (π₯(ballβπ·)π
) β βπ¦ β π π¦ β (π₯(ballβπ·)π
)) |
15 | | simpl1 1192 |
. . . . . . . . . 10
β’ (((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β π· β (βMetβπ)) |
16 | 15 | ad2antrr 725 |
. . . . . . . . 9
β’
(((((π· β
(βMetβπ) β§
πΉ β
(CauFilβπ·) β§
π
β
β+) β§ π β πΉ) β§ π₯ β π) β§ π¦ β π ) β π· β (βMetβπ)) |
17 | | simpll3 1215 |
. . . . . . . . . . 11
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β π
β
β+) |
18 | 17 | rpxrd 12966 |
. . . . . . . . . 10
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β π
β
β*) |
19 | 18 | adantr 482 |
. . . . . . . . 9
β’
(((((π· β
(βMetβπ) β§
πΉ β
(CauFilβπ·) β§
π
β
β+) β§ π β πΉ) β§ π₯ β π) β§ π¦ β π ) β π
β
β*) |
20 | | simplr 768 |
. . . . . . . . 9
β’
(((((π· β
(βMetβπ) β§
πΉ β
(CauFilβπ·) β§
π
β
β+) β§ π β πΉ) β§ π₯ β π) β§ π¦ β π ) β π₯ β π) |
21 | 11 | adantr 482 |
. . . . . . . . . 10
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β π β π) |
22 | 21 | sselda 3948 |
. . . . . . . . 9
β’
(((((π· β
(βMetβπ) β§
πΉ β
(CauFilβπ·) β§
π
β
β+) β§ π β πΉ) β§ π₯ β π) β§ π¦ β π ) β π¦ β π) |
23 | | elbl2 23766 |
. . . . . . . . 9
β’ (((π· β (βMetβπ) β§ π
β β*) β§ (π₯ β π β§ π¦ β π)) β (π¦ β (π₯(ballβπ·)π
) β (π₯π·π¦) < π
)) |
24 | 16, 19, 20, 22, 23 | syl22anc 838 |
. . . . . . . 8
β’
(((((π· β
(βMetβπ) β§
πΉ β
(CauFilβπ·) β§
π
β
β+) β§ π β πΉ) β§ π₯ β π) β§ π¦ β π ) β (π¦ β (π₯(ballβπ·)π
) β (π₯π·π¦) < π
)) |
25 | 24 | ralbidva 3169 |
. . . . . . 7
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β (βπ¦ β π π¦ β (π₯(ballβπ·)π
) β βπ¦ β π (π₯π·π¦) < π
)) |
26 | 14, 25 | bitrid 283 |
. . . . . 6
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β (π β (π₯(ballβπ·)π
) β βπ¦ β π (π₯π·π¦) < π
)) |
27 | 4 | ad2antrr 725 |
. . . . . . 7
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β πΉ β (Filβπ)) |
28 | | simplr 768 |
. . . . . . 7
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β π β πΉ) |
29 | 15 | adantr 482 |
. . . . . . . 8
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β π· β (βMetβπ)) |
30 | | simpr 486 |
. . . . . . . 8
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β π₯ β π) |
31 | | blssm 23794 |
. . . . . . . 8
β’ ((π· β (βMetβπ) β§ π₯ β π β§ π
β β*) β (π₯(ballβπ·)π
) β π) |
32 | 29, 30, 18, 31 | syl3anc 1372 |
. . . . . . 7
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β (π₯(ballβπ·)π
) β π) |
33 | | filss 23227 |
. . . . . . . 8
β’ ((πΉ β (Filβπ) β§ (π β πΉ β§ (π₯(ballβπ·)π
) β π β§ π β (π₯(ballβπ·)π
))) β (π₯(ballβπ·)π
) β πΉ) |
34 | 33 | 3exp2 1355 |
. . . . . . 7
β’ (πΉ β (Filβπ) β (π β πΉ β ((π₯(ballβπ·)π
) β π β (π β (π₯(ballβπ·)π
) β (π₯(ballβπ·)π
) β πΉ)))) |
35 | 27, 28, 32, 34 | syl3c 66 |
. . . . . 6
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β (π β (π₯(ballβπ·)π
) β (π₯(ballβπ·)π
) β πΉ)) |
36 | 26, 35 | sylbird 260 |
. . . . 5
β’ ((((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β§ π₯ β π) β (βπ¦ β π (π₯π·π¦) < π
β (π₯(ballβπ·)π
) β πΉ)) |
37 | 36 | reximdva 3162 |
. . . 4
β’ (((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β (βπ₯ β π βπ¦ β π (π₯π·π¦) < π
β βπ₯ β π (π₯(ballβπ·)π
) β πΉ)) |
38 | 9, 13, 37 | 3syld 60 |
. . 3
β’ (((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β§ π β πΉ) β (βπ₯ β π βπ¦ β π (π₯π·π¦) < π
β βπ₯ β π (π₯(ballβπ·)π
) β πΉ)) |
39 | 38 | rexlimdva 3149 |
. 2
β’ ((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β
(βπ β πΉ βπ₯ β π βπ¦ β π (π₯π·π¦) < π
β βπ₯ β π (π₯(ballβπ·)π
) β πΉ)) |
40 | 2, 39 | mpd 15 |
1
β’ ((π· β (βMetβπ) β§ πΉ β (CauFilβπ·) β§ π
β β+) β
βπ₯ β π (π₯(ballβπ·)π
) β πΉ) |