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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > polpmapN | Structured version Visualization version GIF version |
Description: The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polpmap.b | β’ π΅ = (BaseβπΎ) |
polpmap.o | β’ β₯ = (ocβπΎ) |
polpmap.m | β’ π = (pmapβπΎ) |
polpmap.p | β’ π = (β₯πβπΎ) |
Ref | Expression |
---|---|
polpmapN | β’ ((πΎ β HL β§ π β π΅) β (πβ(πβπ)) = (πβ( β₯ βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | polpmap.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2732 | . . . 4 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | polpmap.m | . . . 4 β’ π = (pmapβπΎ) | |
4 | 1, 2, 3 | pmapssat 38716 | . . 3 β’ ((πΎ β HL β§ π β π΅) β (πβπ) β (AtomsβπΎ)) |
5 | eqid 2732 | . . . 4 β’ (lubβπΎ) = (lubβπΎ) | |
6 | polpmap.o | . . . 4 β’ β₯ = (ocβπΎ) | |
7 | polpmap.p | . . . 4 β’ π = (β₯πβπΎ) | |
8 | 5, 6, 2, 3, 7 | polval2N 38863 | . . 3 β’ ((πΎ β HL β§ (πβπ) β (AtomsβπΎ)) β (πβ(πβπ)) = (πβ( β₯ β((lubβπΎ)β(πβπ))))) |
9 | 4, 8 | syldan 591 | . 2 β’ ((πΎ β HL β§ π β π΅) β (πβ(πβπ)) = (πβ( β₯ β((lubβπΎ)β(πβπ))))) |
10 | eqid 2732 | . . . . . . 7 β’ (leβπΎ) = (leβπΎ) | |
11 | 1, 10, 2, 3 | pmapval 38714 | . . . . . 6 β’ ((πΎ β HL β§ π β π΅) β (πβπ) = {π β (AtomsβπΎ) β£ π(leβπΎ)π}) |
12 | 11 | fveq2d 6895 | . . . . 5 β’ ((πΎ β HL β§ π β π΅) β ((lubβπΎ)β(πβπ)) = ((lubβπΎ)β{π β (AtomsβπΎ) β£ π(leβπΎ)π})) |
13 | hlomcmat 38321 | . . . . . 6 β’ (πΎ β HL β (πΎ β OML β§ πΎ β CLat β§ πΎ β AtLat)) | |
14 | 1, 10, 5, 2 | atlatmstc 38275 | . . . . . 6 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β AtLat) β§ π β π΅) β ((lubβπΎ)β{π β (AtomsβπΎ) β£ π(leβπΎ)π}) = π) |
15 | 13, 14 | sylan 580 | . . . . 5 β’ ((πΎ β HL β§ π β π΅) β ((lubβπΎ)β{π β (AtomsβπΎ) β£ π(leβπΎ)π}) = π) |
16 | 12, 15 | eqtrd 2772 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β ((lubβπΎ)β(πβπ)) = π) |
17 | 16 | fveq2d 6895 | . . 3 β’ ((πΎ β HL β§ π β π΅) β ( β₯ β((lubβπΎ)β(πβπ))) = ( β₯ βπ)) |
18 | 17 | fveq2d 6895 | . 2 β’ ((πΎ β HL β§ π β π΅) β (πβ( β₯ β((lubβπΎ)β(πβπ)))) = (πβ( β₯ βπ))) |
19 | 9, 18 | eqtrd 2772 | 1 β’ ((πΎ β HL β§ π β π΅) β (πβ(πβπ)) = (πβ( β₯ βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 {crab 3432 β wss 3948 class class class wbr 5148 βcfv 6543 Basecbs 17146 lecple 17206 occoc 17207 lubclub 18264 CLatccla 18453 OMLcoml 38131 Atomscatm 38219 AtLatcal 38220 HLchlt 38306 pmapcpmap 38454 β₯πcpolN 38859 |
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-pow 5363 ax-pr 5427 ax-un 7727 |
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-rmo 3376 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-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 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-riota 7367 df-ov 7414 df-oprab 7415 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-oposet 38132 df-ol 38134 df-oml 38135 df-covers 38222 df-ats 38223 df-atl 38254 df-cvlat 38278 df-hlat 38307 df-pmap 38461 df-polarityN 38860 |
This theorem is referenced by: 2polpmapN 38870 2polvalN 38871 3polN 38873 pmapj2N 38886 pmapocjN 38887 2polatN 38889 poml4N 38910 pmapojoinN 38925 |
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