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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dihvalc | Structured version Visualization version GIF version | ||
| Description: Value of isomorphism H for a lattice πΎ when Β¬ π β€ π. (Contributed by NM, 4-Mar-2014.) |
| Ref | Expression |
|---|---|
| dihval.b | β’ π΅ = (BaseβπΎ) |
| dihval.l | β’ β€ = (leβπΎ) |
| dihval.j | β’ β¨ = (joinβπΎ) |
| dihval.m | β’ β§ = (meetβπΎ) |
| dihval.a | β’ π΄ = (AtomsβπΎ) |
| dihval.h | β’ π» = (LHypβπΎ) |
| dihval.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
| dihval.d | β’ π· = ((DIsoBβπΎ)βπ) |
| dihval.c | β’ πΆ = ((DIsoCβπΎ)βπ) |
| dihval.u | β’ π = ((DVecHβπΎ)βπ) |
| dihval.s | β’ π = (LSubSpβπ) |
| dihval.p | β’ β = (LSSumβπ) |
| Ref | Expression |
|---|---|
| dihvalc | β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β (πΌβπ) = (β©π’ β π βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π β§ π)) = π) β π’ = ((πΆβπ) β (π·β(π β§ π)))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dihval.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
| 2 | dihval.l | . . . 4 β’ β€ = (leβπΎ) | |
| 3 | dihval.j | . . . 4 β’ β¨ = (joinβπΎ) | |
| 4 | dihval.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 5 | dihval.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
| 6 | dihval.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 7 | dihval.i | . . . 4 β’ πΌ = ((DIsoHβπΎ)βπ) | |
| 8 | dihval.d | . . . 4 β’ π· = ((DIsoBβπΎ)βπ) | |
| 9 | dihval.c | . . . 4 β’ πΆ = ((DIsoCβπΎ)βπ) | |
| 10 | dihval.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
| 11 | dihval.s | . . . 4 β’ π = (LSubSpβπ) | |
| 12 | dihval.p | . . . 4 β’ β = (LSSumβπ) | |
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dihval 40091 | . . 3 β’ (((πΎ β π β§ π β π») β§ π β π΅) β (πΌβπ) = if(π β€ π, (π·βπ), (β©π’ β π βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π β§ π)) = π) β π’ = ((πΆβπ) β (π·β(π β§ π))))))) |
| 14 | iffalse 4536 | . . 3 β’ (Β¬ π β€ π β if(π β€ π, (π·βπ), (β©π’ β π βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π β§ π)) = π) β π’ = ((πΆβπ) β (π·β(π β§ π)))))) = (β©π’ β π βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π β§ π)) = π) β π’ = ((πΆβπ) β (π·β(π β§ π)))))) | |
| 15 | 13, 14 | sylan9eq 2792 | . 2 β’ ((((πΎ β π β§ π β π») β§ π β π΅) β§ Β¬ π β€ π) β (πΌβπ) = (β©π’ β π βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π β§ π)) = π) β π’ = ((πΆβπ) β (π·β(π β§ π)))))) |
| 16 | 15 | anasss 467 | 1 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β (πΌβπ) = (β©π’ β π βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π β§ π)) = π) β π’ = ((πΆβπ) β (π·β(π β§ π)))))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 ifcif 4527 class class class wbr 5147 βcfv 6540 β©crio 7360 (class class class)co 7405 Basecbs 17140 lecple 17200 joincjn 18260 meetcmee 18261 LSSumclsm 19496 LSubSpclss 20534 Atomscatm 38121 LHypclh 38843 DVecHcdvh 39937 DIsoBcdib 39997 DIsoCcdic 40031 DIsoHcdih 40087 |
| 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-pr 5426 |
| 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-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-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 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-dih 40088 |
| This theorem is referenced by: dihlsscpre 40093 dihvalcqpre 40094 |
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