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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > poml5N | Structured version Visualization version GIF version |
Description: Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
poml4.a | β’ π΄ = (AtomsβπΎ) |
poml4.p | β’ β₯ = (β₯πβπΎ) |
Ref | Expression |
---|---|
poml5N | β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β (( β₯ β(( β₯ βπ) β© ( β₯ βπ))) β© ( β₯ βπ)) = ( β₯ β( β₯ βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1134 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β πΎ β HL) | |
2 | simp3 1136 | . . . 4 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β π β ( β₯ βπ)) | |
3 | poml4.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
4 | poml4.p | . . . . . 6 β’ β₯ = (β₯πβπΎ) | |
5 | 3, 4 | polssatN 39381 | . . . . 5 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) β π΄) |
6 | 5 | 3adant3 1130 | . . . 4 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β ( β₯ βπ) β π΄) |
7 | 2, 6 | sstrd 3990 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β π β π΄) |
8 | 1, 7, 6 | 3jca 1126 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β (πΎ β HL β§ π β π΄ β§ ( β₯ βπ) β π΄)) |
9 | 3, 4 | 3polN 39389 | . . . 4 β’ ((πΎ β HL β§ π β π΄) β ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ βπ)) |
10 | 9 | 3adant3 1130 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ βπ)) |
11 | 2, 10 | jca 511 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β (π β ( β₯ βπ) β§ ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ βπ))) |
12 | 3, 4 | poml4N 39426 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ ( β₯ βπ) β π΄) β ((π β ( β₯ βπ) β§ ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ βπ)) β (( β₯ β(( β₯ βπ) β© ( β₯ βπ))) β© ( β₯ βπ)) = ( β₯ β( β₯ βπ)))) |
13 | 8, 11, 12 | sylc 65 | 1 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β (( β₯ β(( β₯ βπ) β© ( β₯ βπ))) β© ( β₯ βπ)) = ( β₯ β( β₯ βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β© cin 3946 β wss 3947 βcfv 6548 Atomscatm 38735 HLchlt 38822 β₯πcpolN 39375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-psubsp 38976 df-pmap 38977 df-polarityN 39376 |
This theorem is referenced by: osumcllem3N 39431 |
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