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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > oldmm4 | Structured version Visualization version GIF version |
Description: De Morgan's law for meet in an ortholattice. (chdmm4 31286 analog.) (Contributed by NM, 6-Nov-2011.) |
Ref | Expression |
---|---|
oldmm1.b | β’ π΅ = (BaseβπΎ) |
oldmm1.j | β’ β¨ = (joinβπΎ) |
oldmm1.m | β’ β§ = (meetβπΎ) |
oldmm1.o | β’ β₯ = (ocβπΎ) |
Ref | Expression |
---|---|
oldmm4 | β’ ((πΎ β OL β§ π β π΅ β§ π β π΅) β ( β₯ β(( β₯ βπ) β§ ( β₯ βπ))) = (π β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olop 38595 | . . . . 5 β’ (πΎ β OL β πΎ β OP) | |
2 | oldmm1.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
3 | oldmm1.o | . . . . . 6 β’ β₯ = (ocβπΎ) | |
4 | 2, 3 | opoccl 38575 | . . . . 5 β’ ((πΎ β OP β§ π β π΅) β ( β₯ βπ) β π΅) |
5 | 1, 4 | sylan 579 | . . . 4 β’ ((πΎ β OL β§ π β π΅) β ( β₯ βπ) β π΅) |
6 | 5 | 3adant2 1128 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ π β π΅) β ( β₯ βπ) β π΅) |
7 | oldmm1.j | . . . 4 β’ β¨ = (joinβπΎ) | |
8 | oldmm1.m | . . . 4 β’ β§ = (meetβπΎ) | |
9 | 2, 7, 8, 3 | oldmm2 38599 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ ( β₯ βπ) β π΅) β ( β₯ β(( β₯ βπ) β§ ( β₯ βπ))) = (π β¨ ( β₯ β( β₯ βπ)))) |
10 | 6, 9 | syld3an3 1406 | . 2 β’ ((πΎ β OL β§ π β π΅ β§ π β π΅) β ( β₯ β(( β₯ βπ) β§ ( β₯ βπ))) = (π β¨ ( β₯ β( β₯ βπ)))) |
11 | 2, 3 | opococ 38576 | . . . . 5 β’ ((πΎ β OP β§ π β π΅) β ( β₯ β( β₯ βπ)) = π) |
12 | 1, 11 | sylan 579 | . . . 4 β’ ((πΎ β OL β§ π β π΅) β ( β₯ β( β₯ βπ)) = π) |
13 | 12 | 3adant2 1128 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ π β π΅) β ( β₯ β( β₯ βπ)) = π) |
14 | 13 | oveq2d 7420 | . 2 β’ ((πΎ β OL β§ π β π΅ β§ π β π΅) β (π β¨ ( β₯ β( β₯ βπ))) = (π β¨ π)) |
15 | 10, 14 | eqtrd 2766 | 1 β’ ((πΎ β OL β§ π β π΅ β§ π β π΅) β ( β₯ β(( β₯ βπ) β§ ( β₯ βπ))) = (π β¨ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 Basecbs 17151 occoc 17212 joincjn 18274 meetcmee 18275 OPcops 38553 OLcol 38555 |
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 7721 |
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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18258 df-poset 18276 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-lat 18395 df-oposet 38557 df-ol 38559 |
This theorem is referenced by: oldmj1 38602 omlfh3N 38640 pmapj2N 39311 djhlj 40783 |
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