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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatjrot | Structured version Visualization version GIF version |
Description: Rotate lattice join of 3 classes. Frequently-used special case of latjrot 18453 for atoms. (Contributed by NM, 2-Aug-2012.) |
Ref | Expression |
---|---|
hlatjcom.j | β’ β¨ = (joinβπΎ) |
hlatjcom.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlatjrot | 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 | hlatjcom.j | . . 3 β’ β¨ = (joinβπΎ) | |
2 | hlatjcom.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
3 | 1, 2 | hlatj32 38755 | . 2 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)) |
4 | 1, 2 | hlatjcom 38751 | . . . 4 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π ) = (π β¨ π)) |
5 | 4 | 3adant3r2 1180 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β (π β¨ π ) = (π β¨ π)) |
6 | 5 | oveq1d 7420 | . 2 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)) |
7 | 3, 6 | eqtrd 2766 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 joincjn 18276 Atomscatm 38646 HLchlt 38733 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-lat 18397 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 |
This theorem is referenced by: dalemqrprot 39032 dalemrot 39041 dalemrotyz 39042 dalem11 39058 dalem12 39059 dalem39 39095 dalem58 39114 dalem59 39115 dath2 39121 dalawlem13 39267 |
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