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| Mirrors > Home > MPE Home > Th. List > Mathboxes > latmmdiN | Structured version Visualization version GIF version | ||
| Description: Lattice meet distributes over itself. (inindi 4229 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| olmass.b | β’ π΅ = (BaseβπΎ) |
| olmass.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| latmmdiN | 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 β§ π))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ollat 38725 | . . . . 5 β’ (πΎ β OL β πΎ β Lat) | |
| 2 | 1 | adantr 479 | . . . 4 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Lat) |
| 3 | simpr1 1191 | . . . 4 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 4 | olmass.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 5 | olmass.m | . . . . 5 β’ β§ = (meetβπΎ) | |
| 6 | 4, 5 | latmidm 18475 | . . . 4 β’ ((πΎ β Lat β§ π β π΅) β (π β§ π) = π) |
| 7 | 2, 3, 6 | syl2anc 582 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ π) = π) |
| 8 | 7 | oveq1d 7441 | . 2 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ (π β§ π)) = (π β§ (π β§ π))) |
| 9 | simpl 481 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β OL) | |
| 10 | simpr2 1192 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 11 | simpr3 1193 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 12 | 4, 5 | latm4 38745 | . . 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)) = ((π β§ π) β§ (π β§ π))) |
| 13 | 9, 3, 3, 10, 11, 12 | syl122anc 1376 | . 2 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ (π β§ π)) = ((π β§ π) β§ (π β§ π))) |
| 14 | 8, 13 | eqtr3d 2770 | 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 β§ π))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 Basecbs 17189 meetcmee 18313 Latclat 18432 OLcol 38686 |
| 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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18296 df-poset 18314 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-lat 18433 df-oposet 38688 df-ol 38690 |
| This theorem is referenced by: omlfh1N 38770 |
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