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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 31239 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 2724 | . 2 β’ (leβπΎ) = (leβπΎ) | |
| 3 | ollat 38587 | . . 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 38588 | . . . . 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 38558 | . . . 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 18401 | . . 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 1368 | . 2 β’ ((πΎ β OL β§ π β π΅) β (π β§ 0 ) β π΅) |
| 14 | 1, 2, 11 | latmle2 18426 | . . 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 1368 | . 2 β’ ((πΎ β OL β§ π β π΅) β (π β§ 0 )(leβπΎ) 0 ) |
| 16 | 1, 2, 8 | op0le 38560 | . . . 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 18402 | . . . 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 18427 | . . . 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 1369 | . . 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 709 | . 2 β’ ((πΎ β OL β§ π β π΅) β 0 (leβπΎ)(π β§ 0 )) |
| 23 | 1, 2, 4, 13, 10, 15, 22 | latasymd 18406 | 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 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 (class class class)co 7402 Basecbs 17149 lecple 17209 meetcmee 18273 0.cp0 18384 Latclat 18392 OPcops 38546 OLcol 38548 |
| 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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
| 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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-lat 18393 df-oposet 38550 df-ol 38552 |
| This theorem is referenced by: olm02 38611 omlfh1N 38632 |
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