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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 31375 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 1133 | . . 3 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β πΎ β OML) | |
| 2 | simp2 1134 | . . 3 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 3 | omllat 38625 | . . . 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 18404 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
| 7 | 3, 6 | syl3an1 1160 | . . 3 β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
| 8 | 1, 2, 7 | 3jca 1125 | . 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 2726 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
| 10 | 4, 9, 5 | latlej1 18413 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π(leβπΎ)(π β¨ π)) |
| 11 | 3, 10 | syl3an1 1160 | . 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 38627 | . 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 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 lecple 17213 occoc 17214 joincjn 18276 meetcmee 18277 Latclat 18396 OMLcoml 38558 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-lat 18397 df-oposet 38559 df-ol 38561 df-oml 38562 |
| This theorem is referenced by: (None) |
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