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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 2732 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
3 | pmapssat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | pmapssat.m | . . 3 β’ π = (pmapβπΎ) | |
5 | 1, 2, 3, 4 | pmapval 38714 | . 2 β’ ((πΎ β πΆ β§ π β π΅) β (πβπ) = {π β π΄ β£ π(leβπΎ)π}) |
6 | ssrab2 4077 | . 2 β’ {π β π΄ β£ π(leβπΎ)π} β π΄ | |
7 | 5, 6 | eqsstrdi 4036 | 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 396 = wceq 1541 β wcel 2106 {crab 3432 β wss 3948 class class class wbr 5148 βcfv 6543 Basecbs 17146 lecple 17206 Atomscatm 38219 pmapcpmap 38454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-pmap 38461 |
This theorem is referenced by: pmapssbaN 38717 pmapglb2N 38728 pmapglb2xN 38729 pmapjoin 38809 pmapjat1 38810 pmapjat2 38811 pmapjlln1 38812 hlmod1i 38813 polpmapN 38869 2pmaplubN 38883 pmapj2N 38886 pmapocjN 38887 polatN 38888 pmapsubclN 38903 ispsubcl2N 38904 pl42lem2N 38937 pl42lem3N 38938 |
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