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| Mirrors > Home > MPE Home > Th. List > Mathboxes > omllaw5N | Structured version Visualization version GIF version | ||
| Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 30866 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| omllaw5.b | β’ π΅ = (BaseβπΎ) |
| omllaw5.j | β’ β¨ = (joinβπΎ) |
| omllaw5.m | β’ β§ = (meetβπΎ) |
| omllaw5.o | β’ β₯ = (ocβπΎ) |
| Ref | Expression |
|---|---|
| omllaw5N | β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β (π β¨ (( β₯ βπ) β§ (π β¨ π))) = (π β¨ π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1137 | . . 3 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β πΎ β OML) | |
| 2 | simp2 1138 | . . 3 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 3 | omllat 38112 | . . . 4 β’ (πΎ β OML β πΎ β Lat) | |
| 4 | omllaw5.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 5 | omllaw5.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
| 6 | 4, 5 | latjcl 18392 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
| 7 | 3, 6 | syl3an1 1164 | . . 3 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
| 8 | 1, 2, 7 | 3jca 1129 | . 2 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β (πΎ β OML β§ π β π΅ β§ (π β¨ π) β π΅)) |
| 9 | eqid 2733 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
| 10 | 4, 9, 5 | latlej1 18401 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π(leβπΎ)(π β¨ π)) |
| 11 | 3, 10 | syl3an1 1164 | . 2 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β π(leβπΎ)(π β¨ π)) |
| 12 | omllaw5.m | . . 3 β’ β§ = (meetβπΎ) | |
| 13 | omllaw5.o | . . 3 β’ β₯ = (ocβπΎ) | |
| 14 | 4, 9, 5, 12, 13 | omllaw2N 38114 | . 2 β’ ((πΎ β OML β§ π β π΅ β§ (π β¨ π) β π΅) β (π(leβπΎ)(π β¨ π) β (π β¨ (( β₯ βπ) β§ (π β¨ π))) = (π β¨ π))) |
| 15 | 8, 11, 14 | sylc 65 | 1 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β (π β¨ (( β₯ βπ) β§ (π β¨ π))) = (π β¨ π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 occoc 17205 joincjn 18264 meetcmee 18265 Latclat 18384 OMLcoml 38045 |
| 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-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-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-lat 18385 df-oposet 38046 df-ol 38048 df-oml 38049 |
| This theorem is referenced by: (None) |
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