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| Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod1i1 | Structured version Visualization version GIF version | ||
| Description: Version of modular law pmod1i 39361 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 481 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β πΎ β HL) | |
| 2 | simpr2 1192 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 3 | simpr1 1191 | . . . . 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 2728 | . . . . . 6 β’ (pmapβπΎ) = (pmapβπΎ) | |
| 8 | eqid 2728 | . . . . . 6 β’ (+πβπΎ) = (+πβπΎ) | |
| 9 | 4, 5, 6, 7, 8 | pmapjat2 39367 | . . . . 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 1368 | . . . 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 38801 | . . . . 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 39369 | . . . . 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 1413 | . . . 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 1114 | . 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 2734 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Basecbs 17189 lecple 17249 joincjn 18312 meetcmee 18313 Atomscatm 38775 HLchlt 38862 pmapcpmap 39010 +πcpadd 39308 |
| 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
| 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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-proset 18296 df-poset 18314 df-plt 18331 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-p0 18426 df-lat 18433 df-clat 18500 df-oposet 38688 df-ol 38690 df-oml 38691 df-covers 38778 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-psubsp 39016 df-pmap 39017 df-padd 39309 |
| This theorem is referenced by: atmod1i1m 39371 atmod2i1 39374 atmod3i1 39377 atmod4i1 39379 dalawlem6 39389 dalawlem11 39394 dalawlem12 39395 cdleme11g 39778 cdlemednpq 39812 cdleme20c 39824 cdleme22e 39857 cdleme22eALTN 39858 cdleme35c 39964 |
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