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Mirrors > Home > MPE Home > Th. List > latmcom | Structured version Visualization version GIF version |
Description: The join of a lattice commutes. (Contributed by NM, 6-Nov-2011.) |
Ref | Expression |
---|---|
latmcom.b | β’ π΅ = (BaseβπΎ) |
latmcom.m | β’ β§ = (meetβπΎ) |
Ref | Expression |
---|---|
latmcom | β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5671 | . . . . 5 β’ ((π β π΅ β§ π β π΅) β β¨π, πβ© β (π΅ Γ π΅)) | |
2 | 1 | 3adant1 1131 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β β¨π, πβ© β (π΅ Γ π΅)) |
3 | latmcom.b | . . . . . . 7 β’ π΅ = (BaseβπΎ) | |
4 | eqid 2733 | . . . . . . 7 β’ (joinβπΎ) = (joinβπΎ) | |
5 | latmcom.m | . . . . . . 7 β’ β§ = (meetβπΎ) | |
6 | 3, 4, 5 | islat 18327 | . . . . . 6 β’ (πΎ β Lat β (πΎ β Poset β§ (dom (joinβπΎ) = (π΅ Γ π΅) β§ dom β§ = (π΅ Γ π΅)))) |
7 | simprr 772 | . . . . . 6 β’ ((πΎ β Poset β§ (dom (joinβπΎ) = (π΅ Γ π΅) β§ dom β§ = (π΅ Γ π΅))) β dom β§ = (π΅ Γ π΅)) | |
8 | 6, 7 | sylbi 216 | . . . . 5 β’ (πΎ β Lat β dom β§ = (π΅ Γ π΅)) |
9 | 8 | 3ad2ant1 1134 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β dom β§ = (π΅ Γ π΅)) |
10 | 2, 9 | eleqtrrd 2837 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β β¨π, πβ© β dom β§ ) |
11 | opelxpi 5671 | . . . . . 6 β’ ((π β π΅ β§ π β π΅) β β¨π, πβ© β (π΅ Γ π΅)) | |
12 | 11 | ancoms 460 | . . . . 5 β’ ((π β π΅ β§ π β π΅) β β¨π, πβ© β (π΅ Γ π΅)) |
13 | 12 | 3adant1 1131 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β β¨π, πβ© β (π΅ Γ π΅)) |
14 | 13, 9 | eleqtrrd 2837 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β β¨π, πβ© β dom β§ ) |
15 | 10, 14 | jca 513 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (β¨π, πβ© β dom β§ β§ β¨π, πβ© β dom β§ )) |
16 | latpos 18332 | . . 3 β’ (πΎ β Lat β πΎ β Poset) | |
17 | 3, 5 | meetcom 18298 | . . 3 β’ (((πΎ β Poset β§ π β π΅ β§ π β π΅) β§ (β¨π, πβ© β dom β§ β§ β¨π, πβ© β dom β§ )) β (π β§ π) = (π β§ π)) |
18 | 16, 17 | syl3anl1 1413 | . 2 β’ (((πΎ β Lat β§ π β π΅ β§ π β π΅) β§ (β¨π, πβ© β dom β§ β§ β¨π, πβ© β dom β§ )) β (π β§ π) = (π β§ π)) |
19 | 15, 18 | mpdan 686 | 1 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β¨cop 4593 Γ cxp 5632 dom cdm 5634 βcfv 6497 (class class class)co 7358 Basecbs 17088 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-glb 18241 df-meet 18243 df-lat 18326 |
This theorem is referenced by: latleeqm2 18362 latmlem2 18364 latmlej21 18374 latmlej22 18375 mod2ile 18388 olm12 37736 latm12 37738 latm32 37739 latmrot 37740 olm02 37745 omllaw2N 37752 cmtcomlemN 37756 cmtbr3N 37762 omlfh1N 37766 omlmod1i2N 37768 omlspjN 37769 cvlcvrp 37848 intnatN 37916 cvrexch 37929 cvrat4 37952 2atjm 37954 1cvrat 37985 2at0mat0 38034 dalem4 38174 dalem56 38237 atmod2i1 38370 atmod2i2 38371 llnmod2i2 38372 atmod3i1 38373 atmod3i2 38374 llnexchb2lem 38377 dalawlem3 38382 dalawlem4 38383 dalawlem6 38385 dalawlem9 38388 dalawlem11 38390 dalawlem12 38391 dalawlem15 38394 lhpmcvr 38532 4atexlemc 38578 cdleme20zN 38810 cdleme20d 38821 cdleme20l 38831 cdleme20m 38832 cdlemg12 39159 cdlemg17 39186 cdlemg19 39193 cdlemg44a 39240 dihmeetlem17N 39832 dihmeetlem20N 39835 dihmeetALTN 39836 |
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