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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapssat | Structured version Visualization version GIF version |
Description: The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.) |
Ref | Expression |
---|---|
pmapssat.b | β’ π΅ = (BaseβπΎ) |
pmapssat.a | β’ π΄ = (AtomsβπΎ) |
pmapssat.m | β’ π = (pmapβπΎ) |
Ref | Expression |
---|---|
pmapssat | β’ ((πΎ β πΆ β§ π β π΅) β (πβπ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapssat.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2733 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
3 | pmapssat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | pmapssat.m | . . 3 β’ π = (pmapβπΎ) | |
5 | 1, 2, 3, 4 | pmapval 38628 | . 2 β’ ((πΎ β πΆ β§ π β π΅) β (πβπ) = {π β π΄ β£ π(leβπΎ)π}) |
6 | ssrab2 4078 | . 2 β’ {π β π΄ β£ π(leβπΎ)π} β π΄ | |
7 | 5, 6 | eqsstrdi 4037 | 1 β’ ((πΎ β πΆ β§ π β π΅) β (πβπ) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3433 β wss 3949 class class class wbr 5149 βcfv 6544 Basecbs 17144 lecple 17204 Atomscatm 38133 pmapcpmap 38368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-pmap 38375 |
This theorem is referenced by: pmapssbaN 38631 pmapglb2N 38642 pmapglb2xN 38643 pmapjoin 38723 pmapjat1 38724 pmapjat2 38725 pmapjlln1 38726 hlmod1i 38727 polpmapN 38783 2pmaplubN 38797 pmapj2N 38800 pmapocjN 38801 polatN 38802 pmapsubclN 38817 ispsubcl2N 38818 pl42lem2N 38851 pl42lem3N 38852 |
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