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| Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod1i1 | Structured version Visualization version GIF version | ||
| Description: Version of modular law pmod1i 38719 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 484 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β πΎ β HL) | |
| 2 | simpr2 1196 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 3 | simpr1 1195 | . . . . 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 2733 | . . . . . 6 β’ (pmapβπΎ) = (pmapβπΎ) | |
| 8 | eqid 2733 | . . . . . 6 β’ (+πβπΎ) = (+πβπΎ) | |
| 9 | 4, 5, 6, 7, 8 | pmapjat2 38725 | . . . . 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 1372 | . . . 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 38159 | . . . . 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 38727 | . . . . 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 1417 | . . . 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 693 | . . 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 1118 | . 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 2739 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 meetcmee 18265 Atomscatm 38133 HLchlt 38220 pmapcpmap 38368 +πcpadd 38666 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| 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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-psubsp 38374 df-pmap 38375 df-padd 38667 |
| This theorem is referenced by: atmod1i1m 38729 atmod2i1 38732 atmod3i1 38735 atmod4i1 38737 dalawlem6 38747 dalawlem11 38752 dalawlem12 38753 cdleme11g 39136 cdlemednpq 39170 cdleme20c 39182 cdleme22e 39215 cdleme22eALTN 39216 cdleme35c 39322 |
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