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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pclss2polN | Structured version Visualization version GIF version |
Description: The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclss2pol.a | β’ π΄ = (AtomsβπΎ) |
pclss2pol.o | β’ β₯ = (β₯πβπΎ) |
pclss2pol.c | β’ π = (PClβπΎ) |
Ref | Expression |
---|---|
pclss2polN | β’ ((πΎ β HL β§ π β π΄) β (πβπ) β ( β₯ β( β₯ βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 β’ ((πΎ β HL β§ π β π΄) β πΎ β HL) | |
2 | pclss2pol.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
3 | pclss2pol.o | . . . 4 β’ β₯ = (β₯πβπΎ) | |
4 | 2, 3 | 2polssN 39299 | . . 3 β’ ((πΎ β HL β§ π β π΄) β π β ( β₯ β( β₯ βπ))) |
5 | 2, 3 | polssatN 39292 | . . . 4 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) β π΄) |
6 | 2, 3 | polssatN 39292 | . . . 4 β’ ((πΎ β HL β§ ( β₯ βπ) β π΄) β ( β₯ β( β₯ βπ)) β π΄) |
7 | 5, 6 | syldan 590 | . . 3 β’ ((πΎ β HL β§ π β π΄) β ( β₯ β( β₯ βπ)) β π΄) |
8 | pclss2pol.c | . . . 4 β’ π = (PClβπΎ) | |
9 | 2, 8 | pclssN 39278 | . . 3 β’ ((πΎ β HL β§ π β ( β₯ β( β₯ βπ)) β§ ( β₯ β( β₯ βπ)) β π΄) β (πβπ) β (πβ( β₯ β( β₯ βπ)))) |
10 | 1, 4, 7, 9 | syl3anc 1368 | . 2 β’ ((πΎ β HL β§ π β π΄) β (πβπ) β (πβ( β₯ β( β₯ βπ)))) |
11 | eqid 2726 | . . . . 5 β’ (PSubSpβπΎ) = (PSubSpβπΎ) | |
12 | 2, 11, 3 | polsubN 39291 | . . . 4 β’ ((πΎ β HL β§ ( β₯ βπ) β π΄) β ( β₯ β( β₯ βπ)) β (PSubSpβπΎ)) |
13 | 5, 12 | syldan 590 | . . 3 β’ ((πΎ β HL β§ π β π΄) β ( β₯ β( β₯ βπ)) β (PSubSpβπΎ)) |
14 | 11, 8 | pclidN 39280 | . . 3 β’ ((πΎ β HL β§ ( β₯ β( β₯ βπ)) β (PSubSpβπΎ)) β (πβ( β₯ β( β₯ βπ))) = ( β₯ β( β₯ βπ))) |
15 | 13, 14 | syldan 590 | . 2 β’ ((πΎ β HL β§ π β π΄) β (πβ( β₯ β( β₯ βπ))) = ( β₯ β( β₯ βπ))) |
16 | 10, 15 | sseqtrd 4017 | 1 β’ ((πΎ β HL β§ π β π΄) β (πβπ) β ( β₯ β( β₯ βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3943 βcfv 6537 Atomscatm 38646 HLchlt 38733 PSubSpcpsubsp 38880 PClcpclN 39271 β₯πcpolN 39286 |
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 7722 |
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-rmo 3370 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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-psubsp 38887 df-pmap 38888 df-pclN 39272 df-polarityN 39287 |
This theorem is referenced by: pcl0N 39306 |
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