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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pcl0N | Structured version Visualization version GIF version |
Description: The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pcl0.c | β’ π = (PClβπΎ) |
Ref | Expression |
---|---|
pcl0N | β’ (πΎ β HL β (πββ ) = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4392 | . . . 4 β’ β β (AtomsβπΎ) | |
2 | eqid 2728 | . . . . 5 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | eqid 2728 | . . . . 5 β’ (β₯πβπΎ) = (β₯πβπΎ) | |
4 | pcl0.c | . . . . 5 β’ π = (PClβπΎ) | |
5 | 2, 3, 4 | pclss2polN 39388 | . . . 4 β’ ((πΎ β HL β§ β β (AtomsβπΎ)) β (πββ ) β ((β₯πβπΎ)β((β₯πβπΎ)ββ ))) |
6 | 1, 5 | mpan2 690 | . . 3 β’ (πΎ β HL β (πββ ) β ((β₯πβπΎ)β((β₯πβπΎ)ββ ))) |
7 | 3 | 2pol0N 39378 | . . 3 β’ (πΎ β HL β ((β₯πβπΎ)β((β₯πβπΎ)ββ )) = β ) |
8 | 6, 7 | sseqtrd 4018 | . 2 β’ (πΎ β HL β (πββ ) β β ) |
9 | ss0 4394 | . 2 β’ ((πββ ) β β β (πββ ) = β ) | |
10 | 8, 9 | syl 17 | 1 β’ (πΎ β HL β (πββ ) = β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wss 3945 β c0 4318 βcfv 6542 Atomscatm 38729 HLchlt 38816 PClcpclN 39354 β₯πcpolN 39369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-psubsp 38970 df-pmap 38971 df-pclN 39355 df-polarityN 39370 |
This theorem is referenced by: pcl0bN 39390 pclfinclN 39417 |
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