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| Mirrors > Home > MPE Home > Th. List > latmlej12 | 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 |
|---|---|
| latmlej12 | 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 | . . 3 β’ π΅ = (BaseβπΎ) | |
| 2 | latledi.l | . . 3 β’ β€ = (leβπΎ) | |
| 3 | latledi.j | . . 3 β’ β¨ = (joinβπΎ) | |
| 4 | latledi.m | . . 3 β’ β§ = (meetβπΎ) | |
| 5 | 1, 2, 3, 4 | latmlej11 18432 | . 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)) |
| 6 | 1, 3 | latjcom 18401 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
| 7 | 6 | 3adant3r2 1180 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β¨ π) = (π β¨ π)) |
| 8 | 5, 7 | breqtrd 5164 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5138 βcfv 6533 (class class class)co 7401 Basecbs 17142 lecple 17202 joincjn 18265 meetcmee 18266 Latclat 18385 |
| 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 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
| 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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 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 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-poset 18267 df-lub 18300 df-glb 18301 df-join 18302 df-meet 18303 df-lat 18386 |
| This theorem is referenced by: latmlej22 18435 mod1ile 18447 dalawlem3 39200 dalawlem12 39209 |
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