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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 2731 | . . . 4 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | polpmap.m | . . . 4 β’ π = (pmapβπΎ) | |
4 | 1, 2, 3 | pmapssat 38934 | . . 3 β’ ((πΎ β HL β§ π β π΅) β (πβπ) β (AtomsβπΎ)) |
5 | eqid 2731 | . . . 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 39081 | . . 3 β’ ((πΎ β HL β§ (πβπ) β (AtomsβπΎ)) β (πβ(πβπ)) = (πβ( β₯ β((lubβπΎ)β(πβπ))))) |
9 | 4, 8 | syldan 590 | . 2 β’ ((πΎ β HL β§ π β π΅) β (πβ(πβπ)) = (πβ( β₯ β((lubβπΎ)β(πβπ))))) |
10 | eqid 2731 | . . . . . . 7 β’ (leβπΎ) = (leβπΎ) | |
11 | 1, 10, 2, 3 | pmapval 38932 | . . . . . 6 β’ ((πΎ β HL β§ π β π΅) β (πβπ) = {π β (AtomsβπΎ) β£ π(leβπΎ)π}) |
12 | 11 | fveq2d 6896 | . . . . 5 β’ ((πΎ β HL β§ π β π΅) β ((lubβπΎ)β(πβπ)) = ((lubβπΎ)β{π β (AtomsβπΎ) β£ π(leβπΎ)π})) |
13 | hlomcmat 38539 | . . . . . 6 β’ (πΎ β HL β (πΎ β OML β§ πΎ β CLat β§ πΎ β AtLat)) | |
14 | 1, 10, 5, 2 | atlatmstc 38493 | . . . . . 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 579 | . . . . 5 β’ ((πΎ β HL β§ π β π΅) β ((lubβπΎ)β{π β (AtomsβπΎ) β£ π(leβπΎ)π}) = π) |
16 | 12, 15 | eqtrd 2771 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β ((lubβπΎ)β(πβπ)) = π) |
17 | 16 | fveq2d 6896 | . . 3 β’ ((πΎ β HL β§ π β π΅) β ( β₯ β((lubβπΎ)β(πβπ))) = ( β₯ βπ)) |
18 | 17 | fveq2d 6896 | . 2 β’ ((πΎ β HL β§ π β π΅) β (πβ( β₯ β((lubβπΎ)β(πβπ)))) = (πβ( β₯ βπ))) |
19 | 9, 18 | eqtrd 2771 | 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 395 β§ w3a 1086 = wceq 1540 β wcel 2105 {crab 3431 β wss 3949 class class class wbr 5149 βcfv 6544 Basecbs 17149 lecple 17209 occoc 17210 lubclub 18267 CLatccla 18456 OMLcoml 38349 Atomscatm 38437 AtLatcal 38438 HLchlt 38524 pmapcpmap 38672 β₯πcpolN 39077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 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-riota 7368 df-ov 7415 df-oprab 7416 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-pmap 38679 df-polarityN 39078 |
This theorem is referenced by: 2polpmapN 39088 2polvalN 39089 3polN 39091 pmapj2N 39104 pmapocjN 39105 2polatN 39107 poml4N 39128 pmapojoinN 39143 |
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