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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > polsubN | Structured version Visualization version GIF version |
Description: The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polsubsp.a | β’ π΄ = (AtomsβπΎ) |
polsubsp.s | β’ π = (PSubSpβπΎ) |
polsubsp.p | β’ β₯ = (β₯πβπΎ) |
Ref | Expression |
---|---|
polsubN | β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (lubβπΎ) = (lubβπΎ) | |
2 | eqid 2733 | . . 3 β’ (ocβπΎ) = (ocβπΎ) | |
3 | polsubsp.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | eqid 2733 | . . 3 β’ (pmapβπΎ) = (pmapβπΎ) | |
5 | polsubsp.p | . . 3 β’ β₯ = (β₯πβπΎ) | |
6 | 1, 2, 3, 4, 5 | polval2N 38777 | . 2 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) = ((pmapβπΎ)β((ocβπΎ)β((lubβπΎ)βπ)))) |
7 | hllat 38233 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
8 | 7 | adantr 482 | . . 3 β’ ((πΎ β HL β§ π β π΄) β πΎ β Lat) |
9 | hlop 38232 | . . . . 5 β’ (πΎ β HL β πΎ β OP) | |
10 | 9 | adantr 482 | . . . 4 β’ ((πΎ β HL β§ π β π΄) β πΎ β OP) |
11 | hlclat 38228 | . . . . 5 β’ (πΎ β HL β πΎ β CLat) | |
12 | eqid 2733 | . . . . . . 7 β’ (BaseβπΎ) = (BaseβπΎ) | |
13 | 12, 3 | atssbase 38160 | . . . . . 6 β’ π΄ β (BaseβπΎ) |
14 | sstr 3991 | . . . . . 6 β’ ((π β π΄ β§ π΄ β (BaseβπΎ)) β π β (BaseβπΎ)) | |
15 | 13, 14 | mpan2 690 | . . . . 5 β’ (π β π΄ β π β (BaseβπΎ)) |
16 | 12, 1 | clatlubcl 18456 | . . . . 5 β’ ((πΎ β CLat β§ π β (BaseβπΎ)) β ((lubβπΎ)βπ) β (BaseβπΎ)) |
17 | 11, 15, 16 | syl2an 597 | . . . 4 β’ ((πΎ β HL β§ π β π΄) β ((lubβπΎ)βπ) β (BaseβπΎ)) |
18 | 12, 2 | opoccl 38064 | . . . 4 β’ ((πΎ β OP β§ ((lubβπΎ)βπ) β (BaseβπΎ)) β ((ocβπΎ)β((lubβπΎ)βπ)) β (BaseβπΎ)) |
19 | 10, 17, 18 | syl2anc 585 | . . 3 β’ ((πΎ β HL β§ π β π΄) β ((ocβπΎ)β((lubβπΎ)βπ)) β (BaseβπΎ)) |
20 | polsubsp.s | . . . 4 β’ π = (PSubSpβπΎ) | |
21 | 12, 20, 4 | pmapsub 38639 | . . 3 β’ ((πΎ β Lat β§ ((ocβπΎ)β((lubβπΎ)βπ)) β (BaseβπΎ)) β ((pmapβπΎ)β((ocβπΎ)β((lubβπΎ)βπ))) β π) |
22 | 8, 19, 21 | syl2anc 585 | . 2 β’ ((πΎ β HL β§ π β π΄) β ((pmapβπΎ)β((ocβπΎ)β((lubβπΎ)βπ))) β π) |
23 | 6, 22 | eqeltrd 2834 | 1 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3949 βcfv 6544 Basecbs 17144 occoc 17205 lubclub 18262 Latclat 18384 CLatccla 18451 OPcops 38042 Atomscatm 38133 HLchlt 38220 PSubSpcpsubsp 38367 pmapcpmap 38368 β₯πcpolN 38773 |
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-pow 5364 ax-pr 5428 ax-un 7725 |
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-rmo 3377 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-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 7365 df-ov 7412 df-oprab 7413 df-proset 18248 df-poset 18266 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p1 18379 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-psubsp 38374 df-pmap 38375 df-polarityN 38774 |
This theorem is referenced by: polssatN 38779 pclss2polN 38792 psubclsubN 38811 osumcllem1N 38827 |
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