Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolnleat | Structured version Visualization version GIF version | ||
| Description: An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.) |
| Ref | Expression |
|---|---|
| lvolnleat.l | β’ β€ = (leβπΎ) |
| lvolnleat.a | β’ π΄ = (AtomsβπΎ) |
| lvolnleat.v | β’ π = (LVolsβπΎ) |
| Ref | Expression |
|---|---|
| lvolnleat | β’ ((πΎ β HL β§ π β π β§ π β π΄) β Β¬ π β€ π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3simpa 1147 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π΄) β (πΎ β HL β§ π β π)) | |
| 2 | simp3 1137 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π΄) β π β π΄) | |
| 3 | lvolnleat.l | . . . 4 β’ β€ = (leβπΎ) | |
| 4 | eqid 2731 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
| 5 | lvolnleat.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
| 6 | lvolnleat.v | . . . 4 β’ π = (LVolsβπΎ) | |
| 7 | 3, 4, 5, 6 | lvolnle3at 38757 | . . 3 β’ (((πΎ β HL β§ π β π) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β Β¬ π β€ ((π(joinβπΎ)π)(joinβπΎ)π)) |
| 8 | 1, 2, 2, 2, 7 | syl13anc 1371 | . 2 β’ ((πΎ β HL β§ π β π β§ π β π΄) β Β¬ π β€ ((π(joinβπΎ)π)(joinβπΎ)π)) |
| 9 | 4, 5 | hlatjidm 38543 | . . . . . 6 β’ ((πΎ β HL β§ π β π΄) β (π(joinβπΎ)π) = π) |
| 10 | 9 | 3adant2 1130 | . . . . 5 β’ ((πΎ β HL β§ π β π β§ π β π΄) β (π(joinβπΎ)π) = π) |
| 11 | 10 | oveq1d 7427 | . . . 4 β’ ((πΎ β HL β§ π β π β§ π β π΄) β ((π(joinβπΎ)π)(joinβπΎ)π) = (π(joinβπΎ)π)) |
| 12 | 11, 10 | eqtrd 2771 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π΄) β ((π(joinβπΎ)π)(joinβπΎ)π) = π) |
| 13 | 12 | breq2d 5160 | . 2 β’ ((πΎ β HL β§ π β π β§ π β π΄) β (π β€ ((π(joinβπΎ)π)(joinβπΎ)π) β π β€ π)) |
| 14 | 8, 13 | mtbid 324 | 1 β’ ((πΎ β HL β§ π β π β§ π β π΄) β Β¬ π β€ π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 class class class wbr 5148 βcfv 6543 (class class class)co 7412 lecple 17209 joincjn 18269 Atomscatm 38437 HLchlt 38524 LVolsclvol 38668 |
| 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-pow 5363 ax-pr 5427 ax-un 7729 |
| 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-rmo 3375 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-pw 4604 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-oprab 7416 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-lat 18390 df-clat 18457 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 |
| This theorem is referenced by: lvolneatN 38763 lvoln0N 38766 lplncvrlvol 38791 |
| Copyright terms: Public domain | W3C validator |