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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 5072 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
2 | 1 | 3adant1 1072 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
3 | latmcom.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
4 | eqid 2610 | . . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | latmcom.m | . . . . . . 7 ⊢ ∧ = (meet‘𝐾) | |
6 | 3, 4, 5 | islat 16870 | . . . . . 6 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
7 | simprr 792 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (dom (join‘𝐾) = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))) → dom ∧ = (𝐵 × 𝐵)) | |
8 | 6, 7 | sylbi 206 | . . . . 5 ⊢ (𝐾 ∈ Lat → dom ∧ = (𝐵 × 𝐵)) |
9 | 8 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → dom ∧ = (𝐵 × 𝐵)) |
10 | 2, 9 | eleqtrrd 2691 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
11 | opelxpi 5072 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) | |
12 | 11 | ancoms 468 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
13 | 12 | 3adant1 1072 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
14 | 13, 9 | eleqtrrd 2691 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ dom ∧ ) |
15 | 10, 14 | jca 553 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) |
16 | latpos 16873 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
17 | 3, 5 | meetcom 16855 | . . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
18 | 16, 17 | syl3anl1 1366 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
19 | 15, 18 | mpdan 699 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Posetcpo 16763 joincjn 16767 meetcmee 16768 Latclat 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-glb 16798 df-meet 16800 df-lat 16869 |
This theorem is referenced by: latleeqm2 16903 latmlem2 16905 latmlej21 16915 latmlej22 16916 mod2ile 16929 olm12 33533 latm12 33535 latm32 33536 latmrot 33537 olm02 33542 omllaw2N 33549 cmtcomlemN 33553 cmtbr3N 33559 omlfh1N 33563 omlmod1i2N 33565 omlspjN 33566 cvlcvrp 33645 intnatN 33711 cvrexch 33724 cvrat4 33747 2atjm 33749 1cvrat 33780 2at0mat0 33829 dalem4 33969 dalem56 34032 atmod2i1 34165 atmod2i2 34166 llnmod2i2 34167 atmod3i1 34168 atmod3i2 34169 llnexchb2lem 34172 dalawlem3 34177 dalawlem4 34178 dalawlem6 34180 dalawlem9 34183 dalawlem11 34185 dalawlem12 34186 dalawlem15 34189 lhpmcvr 34327 4atexlemc 34373 cdleme20zN 34606 cdleme20d 34618 cdleme20l 34628 cdleme20m 34629 cdlemg12 34956 cdlemg17 34983 cdlemg19 34990 cdlemg44a 35037 dihmeetlem17N 35630 dihmeetlem20N 35633 dihmeetALTN 35634 |
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