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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 30731 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 2732 | . 2 β’ (leβπΎ) = (leβπΎ) | |
| 3 | ollat 38071 | . . 3 β’ (πΎ β OL β πΎ β Lat) | |
| 4 | 3 | adantr 481 | . 2 β’ ((πΎ β OL β§ π β π΅) β πΎ β Lat) |
| 5 | simpr 485 | . . 3 β’ ((πΎ β OL β§ π β π΅) β π β π΅) | |
| 6 | olop 38072 | . . . . 5 β’ (πΎ β OL β πΎ β OP) | |
| 7 | 6 | adantr 481 | . . . 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 38042 | . . . 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 18389 | . . 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 1371 | . 2 β’ ((πΎ β OL β§ π β π΅) β (π β§ 0 ) β π΅) |
| 14 | 1, 2, 11 | latmle2 18414 | . . 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 1371 | . 2 β’ ((πΎ β OL β§ π β π΅) β (π β§ 0 )(leβπΎ) 0 ) |
| 16 | 1, 2, 8 | op0le 38044 | . . . 4 β’ ((πΎ β OP β§ π β π΅) β 0 (leβπΎ)π) |
| 17 | 6, 16 | sylan 580 | . . 3 β’ ((πΎ β OL β§ π β π΅) β 0 (leβπΎ)π) |
| 18 | 1, 2 | latref 18390 | . . . 4 β’ ((πΎ β Lat β§ 0 β π΅) β 0 (leβπΎ) 0 ) |
| 19 | 4, 10, 18 | syl2anc 584 | . . 3 β’ ((πΎ β OL β§ π β π΅) β 0 (leβπΎ) 0 ) |
| 20 | 1, 2, 11 | latlem12 18415 | . . . 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 1372 | . . 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 710 | . 2 β’ ((πΎ β OL β§ π β π΅) β 0 (leβπΎ)(π β§ 0 )) |
| 23 | 1, 2, 4, 13, 10, 15, 22 | latasymd 18394 | 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 396 = wceq 1541 β wcel 2106 class class class wbr 5147 βcfv 6540 (class class class)co 7405 Basecbs 17140 lecple 17200 meetcmee 18261 0.cp0 18372 Latclat 18380 OPcops 38030 OLcol 38032 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18244 df-poset 18262 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-lat 18381 df-oposet 38034 df-ol 38036 |
| This theorem is referenced by: olm02 38095 omlfh1N 38116 |
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