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Mirrors > Home > MPE Home > Th. List > latjle12 | Structured version Visualization version GIF version |
Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 30493 analog.) (Contributed by NM, 17-Sep-2011.) |
Ref | Expression |
---|---|
latlej.b | β’ π΅ = (BaseβπΎ) |
latlej.l | β’ β€ = (leβπΎ) |
latlej.j | β’ β¨ = (joinβπΎ) |
Ref | Expression |
---|---|
latjle12 | 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)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latlej.b | . 2 β’ π΅ = (BaseβπΎ) | |
2 | latlej.l | . 2 β’ β€ = (leβπΎ) | |
3 | latlej.j | . 2 β’ β¨ = (joinβπΎ) | |
4 | latpos 18332 | . . 3 β’ (πΎ β Lat β πΎ β Poset) | |
5 | 4 | adantr 482 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Poset) |
6 | simpr1 1195 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
7 | simpr2 1196 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
8 | simpr3 1197 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
9 | eqid 2733 | . . . 4 β’ (meetβπΎ) = (meetβπΎ) | |
10 | simpl 484 | . . . 4 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Lat) | |
11 | 1, 3, 9, 10, 6, 7 | latcl2 18330 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (β¨π, πβ© β dom β¨ β§ β¨π, πβ© β dom (meetβπΎ))) |
12 | 11 | simpld 496 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β β¨π, πβ© β dom β¨ ) |
13 | 1, 2, 3, 5, 6, 7, 8, 12 | joinle 18280 | 1 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)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β¨cop 4593 class class class wbr 5106 dom cdm 5634 βcfv 6497 (class class class)co 7358 Basecbs 17088 lecple 17145 Posetcpo 18201 joincjn 18205 meetcmee 18206 Latclat 18325 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-poset 18207 df-lub 18240 df-join 18242 df-lat 18326 |
This theorem is referenced by: latleeqj1 18345 latjlej1 18347 latjidm 18356 latledi 18371 latjass 18377 mod1ile 18387 lubun 18409 oldmm1 37725 olj01 37733 cvlexchb1 37838 cvlcvr1 37847 hlrelat 37911 hlrelat2 37912 exatleN 37913 hlrelat3 37921 cvrexchlem 37928 cvratlem 37930 cvrat 37931 atlelt 37947 ps-1 37986 hlatexch3N 37989 hlatexch4 37990 3atlem1 37992 3atlem2 37993 lplnexllnN 38073 2llnjaN 38075 4atlem3 38105 4atlem10 38115 4atlem11b 38117 4atlem11 38118 4atlem12b 38120 4atlem12 38121 2lplnja 38128 dalem1 38168 dalem3 38173 dalem8 38179 dalem16 38188 dalem17 38189 dalem21 38203 dalem25 38207 dalem39 38220 dalem54 38235 dalem60 38241 linepsubN 38261 pmapsub 38277 lneq2at 38287 2llnma3r 38297 cdlema1N 38300 cdlemblem 38302 paddasslem5 38333 paddasslem12 38340 paddasslem13 38341 llnexchb2 38378 dalawlem3 38382 dalawlem5 38384 dalawlem8 38387 dalawlem11 38390 dalawlem12 38391 lhp2lt 38510 lhpexle2lem 38518 lhpexle3lem 38520 4atexlemtlw 38576 4atexlemnclw 38579 lautj 38602 cdlemd3 38709 cdleme3g 38743 cdleme3h 38744 cdleme7d 38755 cdleme11c 38770 cdleme15d 38786 cdleme17b 38796 cdleme19a 38812 cdleme20j 38827 cdleme21c 38836 cdleme22b 38850 cdleme22d 38852 cdleme28a 38879 cdleme35a 38957 cdleme35fnpq 38958 cdleme35b 38959 cdleme35f 38963 cdleme42c 38981 cdleme42i 38992 cdlemf1 39070 cdlemg4c 39121 cdlemg6c 39129 cdlemg8b 39137 cdlemg10 39150 cdlemg11b 39151 cdlemg13a 39160 cdlemg17a 39170 cdlemg18b 39188 cdlemg27a 39201 cdlemg33b0 39210 cdlemg35 39222 cdlemg42 39238 cdlemg46 39244 trljco 39249 tendopltp 39289 cdlemk3 39342 cdlemk10 39352 cdlemk1u 39368 cdlemk39 39425 dialss 39555 dia2dimlem1 39573 dia2dimlem10 39582 dia2dimlem12 39584 cdlemm10N 39627 djajN 39646 diblss 39679 cdlemn2 39704 dihord2pre2 39735 dib2dim 39752 dih2dimb 39753 dih2dimbALTN 39754 dihmeetlem6 39818 dihjatcclem1 39927 |
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