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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod1i2 | Structured version Visualization version GIF version | ||
| Description: Version of modular law pmod1i 39232 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 482 | . . . . 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 2726 | . . . . . 6 β’ (pmapβπΎ) = (pmapβπΎ) | |
| 8 | eqid 2726 | . . . . . 6 β’ (+πβπΎ) = (+πβπΎ) | |
| 9 | 4, 5, 6, 7, 8 | pmapjat1 39237 | . . . . 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 38672 | . . . . . 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 1193 | . . . . 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 39240 | . . . . 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 1369 | . . . 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 691 | . . 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 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))) |
| 20 | 19 | eqcomd 2732 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 lecple 17213 joincjn 18276 meetcmee 18277 Atomscatm 38646 HLchlt 38733 pmapcpmap 38881 +πcpadd 39179 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-psubsp 38887 df-pmap 38888 df-padd 39180 |
| This theorem is referenced by: atmod2i2 39246 atmod3i2 39249 atmod4i2 39251 lhpmod2i2 39422 dihmeetlem7N 40694 dihjatc1 40695 |
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