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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 4187 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 37678 | . . . . 5 β’ (πΎ β OL β πΎ β Lat) | |
2 | 1 | adantr 482 | . . . 4 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Lat) |
3 | simpr1 1195 | . . . 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 18364 | . . . 4 β’ ((πΎ β Lat β§ π β π΅) β (π β§ π) = π) |
7 | 2, 3, 6 | syl2anc 585 | . . 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 7373 | . 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 484 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β OL) | |
10 | simpr2 1196 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
11 | simpr3 1197 | . . 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 37698 | . . 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 1380 | . 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 2779 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17084 meetcmee 18202 Latclat 18321 OLcol 37639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18185 df-poset 18203 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-lat 18322 df-oposet 37641 df-ol 37643 |
This theorem is referenced by: omlfh1N 37723 |
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