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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapsubclN | Structured version Visualization version GIF version |
Description: A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pmapsubcl.b | β’ π΅ = (BaseβπΎ) |
pmapsubcl.m | β’ π = (pmapβπΎ) |
pmapsubcl.c | β’ πΆ = (PSubClβπΎ) |
Ref | Expression |
---|---|
pmapsubclN | β’ ((πΎ β HL β§ π β π΅) β (πβπ) β πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapsubcl.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2725 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | pmapsubcl.m | . . 3 β’ π = (pmapβπΎ) | |
4 | 1, 2, 3 | pmapssat 39287 | . 2 β’ ((πΎ β HL β§ π β π΅) β (πβπ) β (AtomsβπΎ)) |
5 | eqid 2725 | . . 3 β’ (β₯πβπΎ) = (β₯πβπΎ) | |
6 | 1, 3, 5 | 2polpmapN 39441 | . 2 β’ ((πΎ β HL β§ π β π΅) β ((β₯πβπΎ)β((β₯πβπΎ)β(πβπ))) = (πβπ)) |
7 | pmapsubcl.c | . . . 4 β’ πΆ = (PSubClβπΎ) | |
8 | 2, 5, 7 | ispsubclN 39465 | . . 3 β’ (πΎ β HL β ((πβπ) β πΆ β ((πβπ) β (AtomsβπΎ) β§ ((β₯πβπΎ)β((β₯πβπΎ)β(πβπ))) = (πβπ)))) |
9 | 8 | adantr 479 | . 2 β’ ((πΎ β HL β§ π β π΅) β ((πβπ) β πΆ β ((πβπ) β (AtomsβπΎ) β§ ((β₯πβπΎ)β((β₯πβπΎ)β(πβπ))) = (πβπ)))) |
10 | 4, 6, 9 | mpbir2and 711 | 1 β’ ((πΎ β HL β§ π β π΅) β (πβπ) β πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3940 βcfv 6542 Basecbs 17177 Atomscatm 38790 HLchlt 38877 pmapcpmap 39025 β₯πcpolN 39430 PSubClcpscN 39462 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 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 7371 df-ov 7418 df-oprab 7419 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-pmap 39032 df-polarityN 39431 df-psubclN 39463 |
This theorem is referenced by: psubclinN 39476 paddatclN 39477 linepsubclN 39479 polsubclN 39480 pmapojoinN 39496 |
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