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| Mirrors > Home > MPE Home > Th. List > latnlemlt | Structured version Visualization version GIF version | ||
| Description: Negation of "less than or equal to" expressed in terms of meet and less-than. (nssinpss 4257 analog.) (Contributed by NM, 5-Feb-2012.) |
| Ref | Expression |
|---|---|
| latnlemlt.b | β’ π΅ = (BaseβπΎ) |
| latnlemlt.l | β’ β€ = (leβπΎ) |
| latnlemlt.s | β’ < = (ltβπΎ) |
| latnlemlt.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| latnlemlt | β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (Β¬ π β€ π β (π β§ π) < π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latnlemlt.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
| 2 | latnlemlt.l | . . . 4 β’ β€ = (leβπΎ) | |
| 3 | latnlemlt.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 4 | 1, 2, 3 | latmle1 18417 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) β€ π) |
| 5 | 4 | biantrurd 534 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β§ π) β π β ((π β§ π) β€ π β§ (π β§ π) β π))) |
| 6 | 1, 2, 3 | latleeqm1 18420 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β€ π β (π β§ π) = π)) |
| 7 | 6 | necon3bbid 2979 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (Β¬ π β€ π β (π β§ π) β π)) |
| 8 | simp1 1137 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β πΎ β Lat) | |
| 9 | 1, 3 | latmcl 18393 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) β π΅) |
| 10 | simp2 1138 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 11 | latnlemlt.s | . . . 4 β’ < = (ltβπΎ) | |
| 12 | 2, 11 | pltval 18285 | . . 3 β’ ((πΎ β Lat β§ (π β§ π) β π΅ β§ π β π΅) β ((π β§ π) < π β ((π β§ π) β€ π β§ (π β§ π) β π))) |
| 13 | 8, 9, 10, 12 | syl3anc 1372 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β§ π) < π β ((π β§ π) β€ π β§ (π β§ π) β π))) |
| 14 | 5, 7, 13 | 3bitr4d 311 | 1 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (Β¬ π β€ π β (π β§ π) < π)) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 ltcplt 18261 meetcmee 18265 Latclat 18384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-lat 18385 |
| This theorem is referenced by: hlrelat2 38274 |
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