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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > olm01 | Structured version Visualization version GIF version | ||
| Description: Meet with lattice zero is zero. (chm0 31314 analog.) (Contributed by NM, 8-Nov-2011.) |
| Ref | Expression |
|---|---|
| olm0.b | β’ π΅ = (BaseβπΎ) |
| olm0.m | β’ β§ = (meetβπΎ) |
| olm0.z | β’ 0 = (0.βπΎ) |
| Ref | Expression |
|---|---|
| olm01 | β’ ((πΎ β OL β§ π β π΅) β (π β§ 0 ) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | olm0.b | . 2 β’ π΅ = (BaseβπΎ) | |
| 2 | eqid 2728 | . 2 β’ (leβπΎ) = (leβπΎ) | |
| 3 | ollat 38685 | . . 3 β’ (πΎ β OL β πΎ β Lat) | |
| 4 | 3 | adantr 480 | . 2 β’ ((πΎ β OL β§ π β π΅) β πΎ β Lat) |
| 5 | simpr 484 | . . 3 β’ ((πΎ β OL β§ π β π΅) β π β π΅) | |
| 6 | olop 38686 | . . . . 5 β’ (πΎ β OL β πΎ β OP) | |
| 7 | 6 | adantr 480 | . . . 4 β’ ((πΎ β OL β§ π β π΅) β πΎ β OP) |
| 8 | olm0.z | . . . . 5 β’ 0 = (0.βπΎ) | |
| 9 | 1, 8 | op0cl 38656 | . . . 4 β’ (πΎ β OP β 0 β π΅) |
| 10 | 7, 9 | syl 17 | . . 3 β’ ((πΎ β OL β§ π β π΅) β 0 β π΅) |
| 11 | olm0.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 12 | 1, 11 | latmcl 18432 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ 0 β π΅) β (π β§ 0 ) β π΅) |
| 13 | 4, 5, 10, 12 | syl3anc 1369 | . 2 β’ ((πΎ β OL β§ π β π΅) β (π β§ 0 ) β π΅) |
| 14 | 1, 2, 11 | latmle2 18457 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ 0 β π΅) β (π β§ 0 )(leβπΎ) 0 ) |
| 15 | 4, 5, 10, 14 | syl3anc 1369 | . 2 β’ ((πΎ β OL β§ π β π΅) β (π β§ 0 )(leβπΎ) 0 ) |
| 16 | 1, 2, 8 | op0le 38658 | . . . 4 β’ ((πΎ β OP β§ π β π΅) β 0 (leβπΎ)π) |
| 17 | 6, 16 | sylan 579 | . . 3 β’ ((πΎ β OL β§ π β π΅) β 0 (leβπΎ)π) |
| 18 | 1, 2 | latref 18433 | . . . 4 β’ ((πΎ β Lat β§ 0 β π΅) β 0 (leβπΎ) 0 ) |
| 19 | 4, 10, 18 | syl2anc 583 | . . 3 β’ ((πΎ β OL β§ π β π΅) β 0 (leβπΎ) 0 ) |
| 20 | 1, 2, 11 | latlem12 18458 | . . . 4 β’ ((πΎ β Lat β§ ( 0 β π΅ β§ π β π΅ β§ 0 β π΅)) β (( 0 (leβπΎ)π β§ 0 (leβπΎ) 0 ) β 0 (leβπΎ)(π β§ 0 ))) |
| 21 | 4, 10, 5, 10, 20 | syl13anc 1370 | . . 3 β’ ((πΎ β OL β§ π β π΅) β (( 0 (leβπΎ)π β§ 0 (leβπΎ) 0 ) β 0 (leβπΎ)(π β§ 0 ))) |
| 22 | 17, 19, 21 | mpbi2and 711 | . 2 β’ ((πΎ β OL β§ π β π΅) β 0 (leβπΎ)(π β§ 0 )) |
| 23 | 1, 2, 4, 13, 10, 15, 22 | latasymd 18437 | 1 β’ ((πΎ β OL β§ π β π΅) β (π β§ 0 ) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 (class class class)co 7420 Basecbs 17180 lecple 17240 meetcmee 18304 0.cp0 18415 Latclat 18423 OPcops 38644 OLcol 38646 |
| 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-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-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-lat 18424 df-oposet 38648 df-ol 38650 |
| This theorem is referenced by: olm02 38709 omlfh1N 38730 |
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