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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlrelat1 | Structured version Visualization version GIF version |
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 32193, with β§ swapped, analog.) (Contributed by NM, 4-Dec-2011.) |
Ref | Expression |
---|---|
hlrelat1.b | β’ π΅ = (BaseβπΎ) |
hlrelat1.l | β’ β€ = (leβπΎ) |
hlrelat1.s | β’ < = (ltβπΎ) |
hlrelat1.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlrelat1 | β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β (π < π β βπ β π΄ (Β¬ π β€ π β§ π β€ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmat 38869 | . 2 β’ (πΎ β HL β (πΎ β OML β§ πΎ β CLat β§ πΎ β AtLat)) | |
2 | hlrelat1.b | . . 3 β’ π΅ = (BaseβπΎ) | |
3 | hlrelat1.l | . . 3 β’ β€ = (leβπΎ) | |
4 | hlrelat1.s | . . 3 β’ < = (ltβπΎ) | |
5 | hlrelat1.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | 2, 3, 4, 5 | atlrelat1 38825 | . 2 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β AtLat) β§ π β π΅ β§ π β π΅) β (π < π β βπ β π΄ (Β¬ π β€ π β§ π β€ π))) |
7 | 1, 6 | syl3an1 1160 | 1 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β (π < π β βπ β π΄ (Β¬ π β€ π β§ π β€ π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3067 class class class wbr 5152 βcfv 6553 Basecbs 17187 lecple 17247 ltcplt 18307 CLatccla 18497 OMLcoml 38679 Atomscatm 38767 AtLatcal 38768 HLchlt 38854 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 |
This theorem is referenced by: hlrelat5N 38906 hlrelat 38907 hl2at 38910 hlrelat3 38917 cvrexchlem 38924 lhpexle3lem 39516 |
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