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| Mirrors > Home > MPE Home > Th. List > isperf2 | Structured version Visualization version GIF version | ||
| Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.) |
| Ref | Expression |
|---|---|
| lpfval.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| isperf2 | β’ (π½ β Perf β (π½ β Top β§ π β ((limPtβπ½)βπ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lpfval.1 | . . 3 β’ π = βͺ π½ | |
| 2 | 1 | isperf 23075 | . 2 β’ (π½ β Perf β (π½ β Top β§ ((limPtβπ½)βπ) = π)) |
| 3 | ssid 4004 | . . . . 5 β’ π β π | |
| 4 | 1 | lpss 23066 | . . . . 5 β’ ((π½ β Top β§ π β π) β ((limPtβπ½)βπ) β π) |
| 5 | 3, 4 | mpan2 689 | . . . 4 β’ (π½ β Top β ((limPtβπ½)βπ) β π) |
| 6 | eqss 3997 | . . . . 5 β’ (((limPtβπ½)βπ) = π β (((limPtβπ½)βπ) β π β§ π β ((limPtβπ½)βπ))) | |
| 7 | 6 | baib 534 | . . . 4 β’ (((limPtβπ½)βπ) β π β (((limPtβπ½)βπ) = π β π β ((limPtβπ½)βπ))) |
| 8 | 5, 7 | syl 17 | . . 3 β’ (π½ β Top β (((limPtβπ½)βπ) = π β π β ((limPtβπ½)βπ))) |
| 9 | 8 | pm5.32i 573 | . 2 β’ ((π½ β Top β§ ((limPtβπ½)βπ) = π) β (π½ β Top β§ π β ((limPtβπ½)βπ))) |
| 10 | 2, 9 | bitri 274 | 1 β’ (π½ β Perf β (π½ β Top β§ π β ((limPtβπ½)βπ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3949 βͺ cuni 4912 βcfv 6553 Topctop 22815 limPtclp 23058 Perfcperf 23059 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-top 22816 df-cld 22943 df-cls 22945 df-lp 23060 df-perf 23061 |
| This theorem is referenced by: isperf3 23077 |
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