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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatjass | Structured version Visualization version GIF version |
Description: Lattice join is associative. Frequently-used special case of latjass 18377 for atoms. (Contributed by NM, 27-Jul-2012.) |
Ref | Expression |
---|---|
hlatjcom.j | β’ β¨ = (joinβπΎ) |
hlatjcom.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlatjass | β’ ((πΎ β HL 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 | hllat 37871 | . . 3 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | adantr 482 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β πΎ β Lat) |
3 | simpr1 1195 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
4 | eqid 2733 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
5 | hlatjcom.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | 4, 5 | atbase 37797 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
7 | 3, 6 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
8 | simpr2 1196 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
9 | 4, 5 | atbase 37797 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
10 | 8, 9 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
11 | simpr3 1197 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
12 | 4, 5 | atbase 37797 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
13 | 11, 12 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
14 | hlatjcom.j | . . 3 β’ β¨ = (joinβπΎ) | |
15 | 4, 14 | latjass 18377 | . 2 β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ))) β ((π β¨ π) β¨ π ) = (π β¨ (π β¨ π ))) |
16 | 2, 7, 10, 13, 15 | syl13anc 1373 | 1 β’ ((πΎ β HL 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17088 joincjn 18205 Latclat 18325 Atomscatm 37771 HLchlt 37858 |
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 2704 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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 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 18189 df-poset 18207 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-lat 18326 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 |
This theorem is referenced by: hlatj12 37879 4noncolr3 37962 3dim3 37978 3atlem1 37992 3atlem2 37993 4atlem4a 38108 dalemply 38163 dalemsly 38164 dalawlem6 38385 dalawlem11 38390 dalawlem12 38391 4atexlemc 38578 cdleme20c 38820 cdleme35b 38959 dia2dimlem2 39574 |
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