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| Mirrors > Home > MPE Home > Th. List > latmlej21 | Structured version Visualization version GIF version | ||
| Description: Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.) |
| Ref | Expression |
|---|---|
| latledi.b | β’ π΅ = (BaseβπΎ) |
| latledi.l | β’ β€ = (leβπΎ) |
| latledi.j | β’ β¨ = (joinβπΎ) |
| latledi.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| latmlej21 | β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ π) β€ (π β¨ π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latledi.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
| 2 | latledi.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 3 | 1, 2 | latmcom 18395 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
| 4 | 3 | 3adant3r3 1184 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ π) = (π β§ π)) |
| 5 | latledi.l | . . 3 β’ β€ = (leβπΎ) | |
| 6 | latledi.j | . . 3 β’ β¨ = (joinβπΎ) | |
| 7 | 1, 5, 6, 2 | latmlej11 18410 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ π) β€ (π β¨ π)) |
| 8 | 4, 7 | eqbrtrrd 5162 | 1 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ π) β€ (π β¨ π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5138 βcfv 6529 (class class class)co 7390 Basecbs 17123 lecple 17183 joincjn 18243 meetcmee 18244 Latclat 18363 |
| 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 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 |
| 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-poset 18245 df-lub 18278 df-glb 18279 df-join 18280 df-meet 18281 df-lat 18364 |
| This theorem is referenced by: dalawlem3 38533 dalawlem6 38536 |
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