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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 40567 | . . 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 4537 | . . 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 2791 | . 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 466 | 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 395 = wceq 1540 β wcel 2105 βwral 3060 ifcif 4528 class class class wbr 5148 βcfv 6543 β©crio 7367 (class class class)co 7412 Basecbs 17151 lecple 17211 joincjn 18274 meetcmee 18275 LSSumclsm 19550 LSubSpclss 20774 Atomscatm 38597 LHypclh 39319 DVecHcdvh 40413 DIsoBcdib 40473 DIsoCcdic 40507 DIsoHcdih 40563 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-dih 40564 |
| This theorem is referenced by: dihlsscpre 40569 dihvalcqpre 40570 |
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