Nearby theorems
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > latm12 | Structured version Visualization version GIF version | ||
| Description: A rearrangement of lattice meet. (in12 4221 analog.) (Contributed by NM, 8-Nov-2011.) |
| Ref | Expression |
|---|---|
| olmass.b | β’ π΅ = (BaseβπΎ) |
| olmass.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| latm12 | β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ (π β§ π)) = (π β§ (π β§ π))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ollat 38685 | . . . . 5 β’ (πΎ β OL β πΎ β Lat) | |
| 2 | 1 | adantr 480 | . . . 4 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Lat) |
| 3 | simpr1 1192 | . . . 4 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 4 | simpr2 1193 | . . . 4 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 5 | olmass.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 6 | olmass.m | . . . . 5 β’ β§ = (meetβπΎ) | |
| 7 | 5, 6 | latmcom 18455 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
| 8 | 2, 3, 4, 7 | syl3anc 1369 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ π) = (π β§ π)) |
| 9 | 8 | oveq1d 7435 | . 2 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = ((π β§ π) β§ π)) |
| 10 | 5, 6 | latmassOLD 38701 | . 2 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = (π β§ (π β§ π))) |
| 11 | simpr3 1194 | . . . 4 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 12 | 4, 3, 11 | 3jca 1126 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β π΅ β§ π β π΅ β§ π β π΅)) |
| 13 | 5, 6 | latmassOLD 38701 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = (π β§ (π β§ π))) |
| 14 | 12, 13 | syldan 590 | . 2 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = (π β§ (π β§ π))) |
| 15 | 9, 10, 14 | 3eqtr3d 2776 | 1 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ (π β§ π)) = (π β§ (π β§ π))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 Basecbs 17180 meetcmee 18304 Latclat 18423 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-lat 18424 df-oposet 38648 df-ol 38650 |
| This theorem is referenced by: latm4 38705 omlfh1N 38730 dalawlem6 39349 |
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