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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvr2N | Structured version Visualization version GIF version |
Description: Less-than and covers equivalence in a Hilbert lattice. (chcv2 32165 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cvr2.b | β’ π΅ = (BaseβπΎ) |
cvr2.s | β’ < = (ltβπΎ) |
cvr2.j | β’ β¨ = (joinβπΎ) |
cvr2.c | β’ πΆ = ( β βπΎ) |
cvr2.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
cvr2N | β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (π < (π β¨ π) β ππΆ(π β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38835 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | 3ad2ant1 1131 | . . 3 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β πΎ β Lat) |
3 | simp2 1135 | . . 3 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β π β π΅) | |
4 | cvr2.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
5 | cvr2.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
6 | 4, 5 | atbase 38761 | . . . 4 β’ (π β π΄ β π β π΅) |
7 | 6 | 3ad2ant3 1133 | . . 3 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β π β π΅) |
8 | eqid 2728 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
9 | cvr2.s | . . . 4 β’ < = (ltβπΎ) | |
10 | cvr2.j | . . . 4 β’ β¨ = (joinβπΎ) | |
11 | 4, 8, 9, 10 | latnle 18464 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (Β¬ π(leβπΎ)π β π < (π β¨ π))) |
12 | 2, 3, 7, 11 | syl3anc 1369 | . 2 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (Β¬ π(leβπΎ)π β π < (π β¨ π))) |
13 | cvr2.c | . . 3 β’ πΆ = ( β βπΎ) | |
14 | 4, 8, 10, 13, 5 | cvr1 38883 | . 2 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (Β¬ π(leβπΎ)π β ππΆ(π β¨ π))) |
15 | 12, 14 | bitr3d 281 | 1 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (π < (π β¨ π) β ππΆ(π β¨ π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 (class class class)co 7420 Basecbs 17179 lecple 17239 ltcplt 18299 joincjn 18302 Latclat 18422 β ccvr 38734 Atomscatm 38735 HLchlt 38822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-lat 18423 df-clat 18490 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 |
This theorem is referenced by: cvrval4N 38887 |
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