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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 | ⊢ 𝐵 = (Base‘𝐾) |
latmcom.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latmcom | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5588 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
2 | 1 | 3adant1 1132 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
3 | latmcom.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
4 | eqid 2737 | . . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | latmcom.m | . . . . . . 7 ⊢ ∧ = (meet‘𝐾) | |
6 | 3, 4, 5 | islat 17939 | . . . . . 6 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
7 | simprr 773 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (dom (join‘𝐾) = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))) → dom ∧ = (𝐵 × 𝐵)) | |
8 | 6, 7 | sylbi 220 | . . . . 5 ⊢ (𝐾 ∈ Lat → dom ∧ = (𝐵 × 𝐵)) |
9 | 8 | 3ad2ant1 1135 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → dom ∧ = (𝐵 × 𝐵)) |
10 | 2, 9 | eleqtrrd 2841 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
11 | opelxpi 5588 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) | |
12 | 11 | ancoms 462 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
13 | 12 | 3adant1 1132 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
14 | 13, 9 | eleqtrrd 2841 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ dom ∧ ) |
15 | 10, 14 | jca 515 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) |
16 | latpos 17944 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
17 | 3, 5 | meetcom 17910 | . . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
18 | 16, 17 | syl3anl1 1414 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
19 | 15, 18 | mpdan 687 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 〈cop 4547 × cxp 5549 dom cdm 5551 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 Posetcpo 17814 joincjn 17818 meetcmee 17819 Latclat 17937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-glb 17853 df-meet 17855 df-lat 17938 |
This theorem is referenced by: latleeqm2 17974 latmlem2 17976 latmlej21 17986 latmlej22 17987 mod2ile 18000 olm12 36979 latm12 36981 latm32 36982 latmrot 36983 olm02 36988 omllaw2N 36995 cmtcomlemN 36999 cmtbr3N 37005 omlfh1N 37009 omlmod1i2N 37011 omlspjN 37012 cvlcvrp 37091 intnatN 37158 cvrexch 37171 cvrat4 37194 2atjm 37196 1cvrat 37227 2at0mat0 37276 dalem4 37416 dalem56 37479 atmod2i1 37612 atmod2i2 37613 llnmod2i2 37614 atmod3i1 37615 atmod3i2 37616 llnexchb2lem 37619 dalawlem3 37624 dalawlem4 37625 dalawlem6 37627 dalawlem9 37630 dalawlem11 37632 dalawlem12 37633 dalawlem15 37636 lhpmcvr 37774 4atexlemc 37820 cdleme20zN 38052 cdleme20d 38063 cdleme20l 38073 cdleme20m 38074 cdlemg12 38401 cdlemg17 38428 cdlemg19 38435 cdlemg44a 38482 dihmeetlem17N 39074 dihmeetlem20N 39077 dihmeetALTN 39078 |
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