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Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod4i1 | Structured version Visualization version GIF version |
Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
atmod.b | β’ π΅ = (BaseβπΎ) |
atmod.l | β’ β€ = (leβπΎ) |
atmod.j | β’ β¨ = (joinβπΎ) |
atmod.m | β’ β§ = (meetβπΎ) |
atmod.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atmod4i1 | 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) = ((π β¨ π) β§ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38863 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | 3ad2ant1 1130 | . . 3 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 | simp22 1204 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅) β§ π β€ π) β π β π΅) | |
4 | simp23 1205 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅) β§ π β€ π) β π β π΅) | |
5 | atmod.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
6 | atmod.m | . . . . 5 β’ β§ = (meetβπΎ) | |
7 | 5, 6 | latmcl 18429 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) β π΅) |
8 | 2, 3, 4, 7 | syl3anc 1368 | . . 3 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΅) |
9 | simp21 1203 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅) β§ π β€ π) β π β π΄) | |
10 | atmod.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
11 | 5, 10 | atbase 38789 | . . . 4 β’ (π β π΄ β π β π΅) |
12 | 9, 11 | syl 17 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅) β§ π β€ π) β π β π΅) |
13 | atmod.j | . . . 4 β’ β¨ = (joinβπΎ) | |
14 | 5, 13 | latjcom 18436 | . . 3 β’ ((πΎ β Lat β§ (π β§ π) β π΅ β§ π β π΅) β ((π β§ π) β¨ π) = (π β¨ (π β§ π))) |
15 | 2, 8, 12, 14 | syl3anc 1368 | . 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¨ π) = (π β¨ (π β§ π))) |
16 | atmod.l | . . 3 β’ β€ = (leβπΎ) | |
17 | 5, 16, 13, 6, 10 | atmod1i1 39358 | . 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§ π)) = ((π β¨ π) β§ π)) |
18 | 5, 13 | latjcom 18436 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
19 | 2, 12, 3, 18 | syl3anc 1368 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅) β§ π β€ π) β (π β¨ π) = (π β¨ π)) |
20 | 19 | oveq1d 7429 | . 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§ π) = ((π β¨ π) β§ π)) |
21 | 15, 17, 20 | 3eqtrd 2769 | 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) = ((π β¨ π) β§ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6541 (class class class)co 7414 Basecbs 17177 lecple 17237 joincjn 18300 meetcmee 18301 Latclat 18420 Atomscatm 38763 HLchlt 38850 |
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 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7989 df-2nd 7990 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-lat 18421 df-clat 18488 df-oposet 38676 df-ol 38678 df-oml 38679 df-covers 38766 df-ats 38767 df-atl 38798 df-cvlat 38822 df-hlat 38851 df-psubsp 39004 df-pmap 39005 df-padd 39297 |
This theorem is referenced by: dalawlem3 39374 dalawlem7 39378 dalawlem11 39382 cdleme9 39754 cdleme20aN 39810 cdleme22cN 39843 cdleme22d 39844 cdlemh1 40316 dia2dimlem1 40565 dia2dimlem2 40566 dia2dimlem3 40567 |
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