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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atlem4c | Structured version Visualization version GIF version |
Description: Lemma for 4at 38105. Frequently used associative law. (Contributed by NM, 9-Jul-2012.) |
Ref | Expression |
---|---|
4at.l | β’ β€ = (leβπΎ) |
4at.j | β’ β¨ = (joinβπΎ) |
4at.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
4atlem4c | β’ (((πΎ β HL 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))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄)) β πΎ β HL) | |
2 | 1 | hllatd 37855 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄)) β πΎ β Lat) |
3 | eqid 2737 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | 4at.j | . . . 4 β’ β¨ = (joinβπΎ) | |
5 | 4at.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | 3, 4, 5 | hlatjcl 37858 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β (BaseβπΎ)) |
7 | 6 | adantr 482 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄)) β (π β¨ π) β (BaseβπΎ)) |
8 | 3, 5 | atbase 37780 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
9 | 8 | ad2antrl 727 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
10 | 3, 5 | atbase 37780 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
11 | 10 | ad2antll 728 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
12 | 3, 4 | latj12 18380 | . 2 β’ ((πΎ β Lat β§ ((π β¨ π) β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ))) β ((π β¨ π) β¨ (π β¨ π)) = (π β¨ ((π β¨ π) β¨ π))) |
13 | 2, 7, 9, 11, 12 | 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 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 6501 (class class class)co 7362 Basecbs 17090 lecple 17147 joincjn 18207 Latclat 18327 Atomscatm 37754 HLchlt 37841 |
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 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-proset 18191 df-poset 18209 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-lat 18328 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 |
This theorem is referenced by: 4atlem10a 38096 |
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