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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 23005 | . 2 β’ (π½ β Perf β (π½ β Top β§ ((limPtβπ½)βπ) = π)) |
| 3 | ssid 3999 | . . . . 5 β’ π β π | |
| 4 | 1 | lpss 22996 | . . . . 5 β’ ((π½ β Top β§ π β π) β ((limPtβπ½)βπ) β π) |
| 5 | 3, 4 | mpan2 688 | . . . 4 β’ (π½ β Top β ((limPtβπ½)βπ) β π) |
| 6 | eqss 3992 | . . . . 5 β’ (((limPtβπ½)βπ) = π β (((limPtβπ½)βπ) β π β§ π β ((limPtβπ½)βπ))) | |
| 7 | 6 | baib 535 | . . . 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 574 | . 2 β’ ((π½ β Top β§ ((limPtβπ½)βπ) = π) β (π½ β Top β§ π β ((limPtβπ½)βπ))) |
| 10 | 2, 9 | bitri 275 | 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 395 = wceq 1533 β wcel 2098 β wss 3943 βͺ cuni 4902 βcfv 6536 Topctop 22745 limPtclp 22988 Perfcperf 22989 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-top 22746 df-cld 22873 df-cls 22875 df-lp 22990 df-perf 22991 |
| This theorem is referenced by: isperf3 23007 |
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