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Mirrors > Home > MPE Home > Th. List > latjcom | Structured version Visualization version GIF version |
Description: The join of a lattice commutes. (chjcom 30490 analog.) (Contributed by NM, 16-Sep-2011.) |
Ref | Expression |
---|---|
latjcom.b | β’ π΅ = (BaseβπΎ) |
latjcom.j | β’ β¨ = (joinβπΎ) |
Ref | Expression |
---|---|
latjcom | 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 | latjcom.b | . . . . . . 7 β’ π΅ = (BaseβπΎ) | |
4 | latjcom.j | . . . . . . 7 β’ β¨ = (joinβπΎ) | |
5 | eqid 2733 | . . . . . . 7 β’ (meetβπΎ) = (meetβπΎ) | |
6 | 3, 4, 5 | islat 18327 | . . . . . 6 β’ (πΎ β Lat β (πΎ β Poset β§ (dom β¨ = (π΅ Γ π΅) β§ dom (meetβπΎ) = (π΅ Γ π΅)))) |
7 | simprl 770 | . . . . . 6 β’ ((πΎ β Poset β§ (dom β¨ = (π΅ Γ π΅) β§ dom (meetβπΎ) = (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, 4 | joincom 18296 | . . 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-lub 18240 df-join 18242 df-lat 18326 |
This theorem is referenced by: latleeqj2 18346 latjlej2 18348 latnle 18367 latmlej12 18373 latj12 18378 latj32 18379 latj13 18380 latj31 18381 latj4rot 18384 mod2ile 18388 latdisdlem 18390 olj02 37734 omllaw4 37754 cmt2N 37758 cmtbr3N 37762 cvlexch2 37837 cvlexchb2 37839 cvlatexchb2 37843 cvlatexch2 37845 cvlatexch3 37846 cvlatcvr2 37850 cvlsupr2 37851 cvlsupr7 37856 cvlsupr8 37857 hlatjcom 37876 hlrelat5N 37910 cvrval5 37924 cvrexch 37929 cvratlem 37930 cvrat 37931 2atlt 37948 cvrat3 37951 cvrat4 37952 cvrat42 37953 4noncolr3 37962 1cvrat 37985 3atlem1 37992 4atlem4d 38111 4atlem12 38121 paddcom 38322 paddasslem2 38330 pmapjat2 38363 atmod2i1 38370 atmod2i2 38371 llnmod2i2 38372 atmod4i1 38375 atmod4i2 38376 dalawlem4 38383 dalawlem9 38388 dalawlem12 38391 lhpjat2 38530 lhple 38551 trljat1 38675 trljat2 38676 cdlemc1 38700 cdlemc6 38705 cdlemd1 38707 cdleme5 38749 cdleme9 38762 cdleme10 38763 cdleme19e 38816 trlcolem 39235 trljco2 39250 cdlemk7 39357 cdlemk7u 39379 cdlemkid1 39431 dih1 39795 dihjatc2N 39821 |
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