Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod1i2 | Structured version Visualization version GIF version | ||
| Description: Version of modular law pmod1i 38707 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-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 |
|---|---|
| atmod1i2 | 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 | pmapjat1 38712 | . . . . 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 38147 | . . . . . 6 β’ (π β π΄ β π β π΅) |
| 12 | 3, 11 | syl 17 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β π β π΅) |
| 13 | simpr3 1196 | . . . . 5 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 14 | atmod.l | . . . . . 6 β’ β€ = (leβπΎ) | |
| 15 | atmod.m | . . . . . 6 β’ β§ = (meetβπΎ) | |
| 16 | 4, 14, 5, 15, 7, 8 | hlmod1i 38715 | . . . . 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)))) |
| 17 | 1, 2, 12, 13, 16 | syl13anc 1372 | . . . 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)))) |
| 18 | 10, 17 | 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)))) |
| 19 | 18 | 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))) |
| 20 | 19 | 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 5147 βcfv 6540 (class class class)co 7405 Basecbs 17140 lecple 17200 joincjn 18260 meetcmee 18261 Atomscatm 38121 HLchlt 38208 pmapcpmap 38356 +πcpadd 38654 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
| 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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-lat 18381 df-clat 18448 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-psubsp 38362 df-pmap 38363 df-padd 38655 |
| This theorem is referenced by: atmod2i2 38721 atmod3i2 38724 atmod4i2 38726 lhpmod2i2 38897 dihmeetlem7N 40169 dihjatc1 40170 |
| Copyright terms: Public domain | W3C validator |