Step | Hyp | Ref
| Expression |
1 | | isfcls2 23737 |
. 2
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (π΄ β (π½ fClus πΉ) β βπ β πΉ π΄ β ((clsβπ½)βπ ))) |
2 | | filn0 23586 |
. . . . . 6
β’ (πΉ β (Filβπ) β πΉ β β
) |
3 | 2 | adantl 482 |
. . . . 5
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β πΉ β β
) |
4 | | r19.2z 4494 |
. . . . . 6
β’ ((πΉ β β
β§
βπ β πΉ π΄ β ((clsβπ½)βπ )) β βπ β πΉ π΄ β ((clsβπ½)βπ )) |
5 | 4 | ex 413 |
. . . . 5
β’ (πΉ β β
β
(βπ β πΉ π΄ β ((clsβπ½)βπ ) β βπ β πΉ π΄ β ((clsβπ½)βπ ))) |
6 | 3, 5 | syl 17 |
. . . 4
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (βπ β πΉ π΄ β ((clsβπ½)βπ ) β βπ β πΉ π΄ β ((clsβπ½)βπ ))) |
7 | | topontop 22635 |
. . . . . . . . 9
β’ (π½ β (TopOnβπ) β π½ β Top) |
8 | 7 | ad2antrr 724 |
. . . . . . . 8
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π β πΉ) β π½ β Top) |
9 | | filelss 23576 |
. . . . . . . . . 10
β’ ((πΉ β (Filβπ) β§ π β πΉ) β π β π) |
10 | 9 | adantll 712 |
. . . . . . . . 9
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π β πΉ) β π β π) |
11 | | toponuni 22636 |
. . . . . . . . . 10
β’ (π½ β (TopOnβπ) β π = βͺ π½) |
12 | 11 | ad2antrr 724 |
. . . . . . . . 9
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π β πΉ) β π = βͺ π½) |
13 | 10, 12 | sseqtrd 4022 |
. . . . . . . 8
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π β πΉ) β π β βͺ π½) |
14 | | eqid 2732 |
. . . . . . . . 9
β’ βͺ π½ =
βͺ π½ |
15 | 14 | clsss3 22783 |
. . . . . . . 8
β’ ((π½ β Top β§ π β βͺ π½)
β ((clsβπ½)βπ ) β βͺ π½) |
16 | 8, 13, 15 | syl2anc 584 |
. . . . . . 7
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π β πΉ) β ((clsβπ½)βπ ) β βͺ π½) |
17 | 16, 12 | sseqtrrd 4023 |
. . . . . 6
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π β πΉ) β ((clsβπ½)βπ ) β π) |
18 | 17 | sseld 3981 |
. . . . 5
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π β πΉ) β (π΄ β ((clsβπ½)βπ ) β π΄ β π)) |
19 | 18 | rexlimdva 3155 |
. . . 4
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (βπ β πΉ π΄ β ((clsβπ½)βπ ) β π΄ β π)) |
20 | 6, 19 | syld 47 |
. . 3
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (βπ β πΉ π΄ β ((clsβπ½)βπ ) β π΄ β π)) |
21 | 20 | pm4.71rd 563 |
. 2
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (βπ β πΉ π΄ β ((clsβπ½)βπ ) β (π΄ β π β§ βπ β πΉ π΄ β ((clsβπ½)βπ )))) |
22 | 7 | ad3antrrr 728 |
. . . . . 6
β’ ((((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π΄ β π) β§ π β πΉ) β π½ β Top) |
23 | 13 | adantlr 713 |
. . . . . 6
β’ ((((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π΄ β π) β§ π β πΉ) β π β βͺ π½) |
24 | | simplr 767 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π΄ β π) β§ π β πΉ) β π΄ β π) |
25 | 11 | ad3antrrr 728 |
. . . . . . 7
β’ ((((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π΄ β π) β§ π β πΉ) β π = βͺ π½) |
26 | 24, 25 | eleqtrd 2835 |
. . . . . 6
β’ ((((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π΄ β π) β§ π β πΉ) β π΄ β βͺ π½) |
27 | 14 | elcls 22797 |
. . . . . 6
β’ ((π½ β Top β§ π β βͺ π½
β§ π΄ β βͺ π½)
β (π΄ β
((clsβπ½)βπ ) β βπ β π½ (π΄ β π β (π β© π ) β β
))) |
28 | 22, 23, 26, 27 | syl3anc 1371 |
. . . . 5
β’ ((((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π΄ β π) β§ π β πΉ) β (π΄ β ((clsβπ½)βπ ) β βπ β π½ (π΄ β π β (π β© π ) β β
))) |
29 | 28 | ralbidva 3175 |
. . . 4
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π΄ β π) β (βπ β πΉ π΄ β ((clsβπ½)βπ ) β βπ β πΉ βπ β π½ (π΄ β π β (π β© π ) β β
))) |
30 | | ralcom 3286 |
. . . . 5
β’
(βπ β
πΉ βπ β π½ (π΄ β π β (π β© π ) β β
) β βπ β π½ βπ β πΉ (π΄ β π β (π β© π ) β β
)) |
31 | | r19.21v 3179 |
. . . . . 6
β’
(βπ β
πΉ (π΄ β π β (π β© π ) β β
) β (π΄ β π β βπ β πΉ (π β© π ) β β
)) |
32 | 31 | ralbii 3093 |
. . . . 5
β’
(βπ β
π½ βπ β πΉ (π΄ β π β (π β© π ) β β
) β βπ β π½ (π΄ β π β βπ β πΉ (π β© π ) β β
)) |
33 | 30, 32 | bitri 274 |
. . . 4
β’
(βπ β
πΉ βπ β π½ (π΄ β π β (π β© π ) β β
) β βπ β π½ (π΄ β π β βπ β πΉ (π β© π ) β β
)) |
34 | 29, 33 | bitrdi 286 |
. . 3
β’ (((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β§ π΄ β π) β (βπ β πΉ π΄ β ((clsβπ½)βπ ) β βπ β π½ (π΄ β π β βπ β πΉ (π β© π ) β β
))) |
35 | 34 | pm5.32da 579 |
. 2
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β ((π΄ β π β§ βπ β πΉ π΄ β ((clsβπ½)βπ )) β (π΄ β π β§ βπ β π½ (π΄ β π β βπ β πΉ (π β© π ) β β
)))) |
36 | 1, 21, 35 | 3bitrd 304 |
1
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (π΄ β (π½ fClus πΉ) β (π΄ β π β§ βπ β π½ (π΄ β π β βπ β πΉ (π β© π ) β β
)))) |