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| Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod1i1 | Structured version Visualization version GIF version | ||
| Description: Version of modular law pmod1i 38805 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-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 |
|---|---|
| atmod1i1 | 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 | simpl 483 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β πΎ β HL) | |
| 2 | simpr2 1195 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 3 | simpr1 1194 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β π β π΄) | |
| 4 | atmod.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
| 5 | atmod.j | . . . . . 6 β’ β¨ = (joinβπΎ) | |
| 6 | atmod.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
| 7 | eqid 2732 | . . . . . 6 β’ (pmapβπΎ) = (pmapβπΎ) | |
| 8 | eqid 2732 | . . . . . 6 β’ (+πβπΎ) = (+πβπΎ) | |
| 9 | 4, 5, 6, 7, 8 | pmapjat2 38811 | . . . . 5 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β ((pmapβπΎ)β(π β¨ π)) = (((pmapβπΎ)βπ)(+πβπΎ)((pmapβπΎ)βπ))) |
| 10 | 1, 2, 3, 9 | syl3anc 1371 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β ((pmapβπΎ)β(π β¨ π)) = (((pmapβπΎ)βπ)(+πβπΎ)((pmapβπΎ)βπ))) |
| 11 | 4, 6 | atbase 38245 | . . . . 5 β’ (π β π΄ β π β π΅) |
| 12 | atmod.l | . . . . . 6 β’ β€ = (leβπΎ) | |
| 13 | atmod.m | . . . . . 6 β’ β§ = (meetβπΎ) | |
| 14 | 4, 12, 5, 13, 7, 8 | hlmod1i 38813 | . . . . 5 β’ ((πΎ β HL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β€ π β§ ((pmapβπΎ)β(π β¨ π)) = (((pmapβπΎ)βπ)(+πβπΎ)((pmapβπΎ)βπ))) β ((π β¨ π) β§ π) = (π β¨ (π β§ π)))) |
| 15 | 11, 14 | syl3anr1 1416 | . . . 4 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β ((π β€ π β§ ((pmapβπΎ)β(π β¨ π)) = (((pmapβπΎ)βπ)(+πβπΎ)((pmapβπΎ)βπ))) β ((π β¨ π) β§ π) = (π β¨ (π β§ π)))) |
| 16 | 10, 15 | mpan2d 692 | . . 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) = (π β¨ (π β§ π)))) |
| 17 | 16 | 3impia 1117 | . 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 | 17 | eqcomd 2738 | 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§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7411 Basecbs 17146 lecple 17206 joincjn 18266 meetcmee 18267 Atomscatm 38219 HLchlt 38306 pmapcpmap 38454 +πcpadd 38752 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-lat 18387 df-clat 18454 df-oposet 38132 df-ol 38134 df-oml 38135 df-covers 38222 df-ats 38223 df-atl 38254 df-cvlat 38278 df-hlat 38307 df-psubsp 38460 df-pmap 38461 df-padd 38753 |
| This theorem is referenced by: atmod1i1m 38815 atmod2i1 38818 atmod3i1 38821 atmod4i1 38823 dalawlem6 38833 dalawlem11 38838 dalawlem12 38839 cdleme11g 39222 cdlemednpq 39256 cdleme20c 39268 cdleme22e 39301 cdleme22eALTN 39302 cdleme35c 39408 |
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