Step | Hyp | Ref
| Expression |
1 | | t1top 22704 |
. . . . . . 7
β’ (π½ β Fre β π½ β Top) |
2 | | lpcls.1 |
. . . . . . . . . 10
β’ π = βͺ
π½ |
3 | 2 | clsss3 22433 |
. . . . . . . . 9
β’ ((π½ β Top β§ π β π) β ((clsβπ½)βπ) β π) |
4 | 3 | ssdifssd 4106 |
. . . . . . . 8
β’ ((π½ β Top β§ π β π) β (((clsβπ½)βπ) β {π₯}) β π) |
5 | 2 | clsss3 22433 |
. . . . . . . 8
β’ ((π½ β Top β§
(((clsβπ½)βπ) β {π₯}) β π) β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β π) |
6 | 4, 5 | syldan 592 |
. . . . . . 7
β’ ((π½ β Top β§ π β π) β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β π) |
7 | 1, 6 | sylan 581 |
. . . . . 6
β’ ((π½ β Fre β§ π β π) β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β π) |
8 | 7 | sseld 3947 |
. . . . 5
β’ ((π½ β Fre β§ π β π) β (π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β π₯ β π)) |
9 | | ssdifss 4099 |
. . . . . . . . . . 11
β’ (π β π β (π β {π₯}) β π) |
10 | 2 | clscld 22421 |
. . . . . . . . . . 11
β’ ((π½ β Top β§ (π β {π₯}) β π) β ((clsβπ½)β(π β {π₯})) β (Clsdβπ½)) |
11 | 1, 9, 10 | syl2an 597 |
. . . . . . . . . 10
β’ ((π½ β Fre β§ π β π) β ((clsβπ½)β(π β {π₯})) β (Clsdβπ½)) |
12 | 11 | adantr 482 |
. . . . . . . . 9
β’ (((π½ β Fre β§ π β π) β§ π₯ β π) β ((clsβπ½)β(π β {π₯})) β (Clsdβπ½)) |
13 | 2 | t1sncld 22700 |
. . . . . . . . . . . . 13
β’ ((π½ β Fre β§ π₯ β π) β {π₯} β (Clsdβπ½)) |
14 | 13 | adantlr 714 |
. . . . . . . . . . . 12
β’ (((π½ β Fre β§ π β π) β§ π₯ β π) β {π₯} β (Clsdβπ½)) |
15 | | uncld 22415 |
. . . . . . . . . . . 12
β’ (({π₯} β (Clsdβπ½) β§ ((clsβπ½)β(π β {π₯})) β (Clsdβπ½)) β ({π₯} βͺ ((clsβπ½)β(π β {π₯}))) β (Clsdβπ½)) |
16 | 14, 12, 15 | syl2anc 585 |
. . . . . . . . . . 11
β’ (((π½ β Fre β§ π β π) β§ π₯ β π) β ({π₯} βͺ ((clsβπ½)β(π β {π₯}))) β (Clsdβπ½)) |
17 | 2 | sscls 22430 |
. . . . . . . . . . . . . 14
β’ ((π½ β Top β§ (π β {π₯}) β π) β (π β {π₯}) β ((clsβπ½)β(π β {π₯}))) |
18 | 1, 9, 17 | syl2an 597 |
. . . . . . . . . . . . 13
β’ ((π½ β Fre β§ π β π) β (π β {π₯}) β ((clsβπ½)β(π β {π₯}))) |
19 | | ssundif 4449 |
. . . . . . . . . . . . 13
β’ (π β ({π₯} βͺ ((clsβπ½)β(π β {π₯}))) β (π β {π₯}) β ((clsβπ½)β(π β {π₯}))) |
20 | 18, 19 | sylibr 233 |
. . . . . . . . . . . 12
β’ ((π½ β Fre β§ π β π) β π β ({π₯} βͺ ((clsβπ½)β(π β {π₯})))) |
21 | 20 | adantr 482 |
. . . . . . . . . . 11
β’ (((π½ β Fre β§ π β π) β§ π₯ β π) β π β ({π₯} βͺ ((clsβπ½)β(π β {π₯})))) |
22 | 2 | clsss2 22446 |
. . . . . . . . . . 11
β’ ((({π₯} βͺ ((clsβπ½)β(π β {π₯}))) β (Clsdβπ½) β§ π β ({π₯} βͺ ((clsβπ½)β(π β {π₯})))) β ((clsβπ½)βπ) β ({π₯} βͺ ((clsβπ½)β(π β {π₯})))) |
23 | 16, 21, 22 | syl2anc 585 |
. . . . . . . . . 10
β’ (((π½ β Fre β§ π β π) β§ π₯ β π) β ((clsβπ½)βπ) β ({π₯} βͺ ((clsβπ½)β(π β {π₯})))) |
24 | | ssundif 4449 |
. . . . . . . . . 10
β’
(((clsβπ½)βπ) β ({π₯} βͺ ((clsβπ½)β(π β {π₯}))) β (((clsβπ½)βπ) β {π₯}) β ((clsβπ½)β(π β {π₯}))) |
25 | 23, 24 | sylib 217 |
. . . . . . . . 9
β’ (((π½ β Fre β§ π β π) β§ π₯ β π) β (((clsβπ½)βπ) β {π₯}) β ((clsβπ½)β(π β {π₯}))) |
26 | 2 | clsss2 22446 |
. . . . . . . . 9
β’
((((clsβπ½)β(π β {π₯})) β (Clsdβπ½) β§ (((clsβπ½)βπ) β {π₯}) β ((clsβπ½)β(π β {π₯}))) β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β ((clsβπ½)β(π β {π₯}))) |
27 | 12, 25, 26 | syl2anc 585 |
. . . . . . . 8
β’ (((π½ β Fre β§ π β π) β§ π₯ β π) β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β ((clsβπ½)β(π β {π₯}))) |
28 | 27 | sseld 3947 |
. . . . . . 7
β’ (((π½ β Fre β§ π β π) β§ π₯ β π) β (π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β π₯ β ((clsβπ½)β(π β {π₯})))) |
29 | 28 | ex 414 |
. . . . . 6
β’ ((π½ β Fre β§ π β π) β (π₯ β π β (π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β π₯ β ((clsβπ½)β(π β {π₯}))))) |
30 | 29 | com23 86 |
. . . . 5
β’ ((π½ β Fre β§ π β π) β (π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β (π₯ β π β π₯ β ((clsβπ½)β(π β {π₯}))))) |
31 | 8, 30 | mpdd 43 |
. . . 4
β’ ((π½ β Fre β§ π β π) β (π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β π₯ β ((clsβπ½)β(π β {π₯})))) |
32 | 1 | adantr 482 |
. . . . . 6
β’ ((π½ β Fre β§ π β π) β π½ β Top) |
33 | 1, 3 | sylan 581 |
. . . . . . 7
β’ ((π½ β Fre β§ π β π) β ((clsβπ½)βπ) β π) |
34 | 33 | ssdifssd 4106 |
. . . . . 6
β’ ((π½ β Fre β§ π β π) β (((clsβπ½)βπ) β {π₯}) β π) |
35 | 2 | sscls 22430 |
. . . . . . . 8
β’ ((π½ β Top β§ π β π) β π β ((clsβπ½)βπ)) |
36 | 1, 35 | sylan 581 |
. . . . . . 7
β’ ((π½ β Fre β§ π β π) β π β ((clsβπ½)βπ)) |
37 | 36 | ssdifd 4104 |
. . . . . 6
β’ ((π½ β Fre β§ π β π) β (π β {π₯}) β (((clsβπ½)βπ) β {π₯})) |
38 | 2 | clsss 22428 |
. . . . . 6
β’ ((π½ β Top β§
(((clsβπ½)βπ) β {π₯}) β π β§ (π β {π₯}) β (((clsβπ½)βπ) β {π₯})) β ((clsβπ½)β(π β {π₯})) β ((clsβπ½)β(((clsβπ½)βπ) β {π₯}))) |
39 | 32, 34, 37, 38 | syl3anc 1372 |
. . . . 5
β’ ((π½ β Fre β§ π β π) β ((clsβπ½)β(π β {π₯})) β ((clsβπ½)β(((clsβπ½)βπ) β {π₯}))) |
40 | 39 | sseld 3947 |
. . . 4
β’ ((π½ β Fre β§ π β π) β (π₯ β ((clsβπ½)β(π β {π₯})) β π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})))) |
41 | 31, 40 | impbid 211 |
. . 3
β’ ((π½ β Fre β§ π β π) β (π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})) β π₯ β ((clsβπ½)β(π β {π₯})))) |
42 | 2 | islp 22514 |
. . . . 5
β’ ((π½ β Top β§
((clsβπ½)βπ) β π) β (π₯ β ((limPtβπ½)β((clsβπ½)βπ)) β π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})))) |
43 | 3, 42 | syldan 592 |
. . . 4
β’ ((π½ β Top β§ π β π) β (π₯ β ((limPtβπ½)β((clsβπ½)βπ)) β π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})))) |
44 | 1, 43 | sylan 581 |
. . 3
β’ ((π½ β Fre β§ π β π) β (π₯ β ((limPtβπ½)β((clsβπ½)βπ)) β π₯ β ((clsβπ½)β(((clsβπ½)βπ) β {π₯})))) |
45 | 2 | islp 22514 |
. . . 4
β’ ((π½ β Top β§ π β π) β (π₯ β ((limPtβπ½)βπ) β π₯ β ((clsβπ½)β(π β {π₯})))) |
46 | 1, 45 | sylan 581 |
. . 3
β’ ((π½ β Fre β§ π β π) β (π₯ β ((limPtβπ½)βπ) β π₯ β ((clsβπ½)β(π β {π₯})))) |
47 | 41, 44, 46 | 3bitr4d 311 |
. 2
β’ ((π½ β Fre β§ π β π) β (π₯ β ((limPtβπ½)β((clsβπ½)βπ)) β π₯ β ((limPtβπ½)βπ))) |
48 | 47 | eqrdv 2731 |
1
β’ ((π½ β Fre β§ π β π) β ((limPtβπ½)β((clsβπ½)βπ)) = ((limPtβπ½)βπ)) |