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| 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 5721 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
| 2 | 1 | 3adant1 1130 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | 
| 3 | latmcom.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
| 4 | eqid 2736 | . . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 5 | latmcom.m | . . . . . . 7 ⊢ ∧ = (meet‘𝐾) | |
| 6 | 3, 4, 5 | islat 18479 | . . . . . 6 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) | 
| 7 | simprr 772 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (dom (join‘𝐾) = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))) → dom ∧ = (𝐵 × 𝐵)) | |
| 8 | 6, 7 | sylbi 217 | . . . . 5 ⊢ (𝐾 ∈ Lat → dom ∧ = (𝐵 × 𝐵)) | 
| 9 | 8 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → dom ∧ = (𝐵 × 𝐵)) | 
| 10 | 2, 9 | eleqtrrd 2843 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) | 
| 11 | opelxpi 5721 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) | |
| 12 | 11 | ancoms 458 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) | 
| 13 | 12 | 3adant1 1130 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) | 
| 14 | 13, 9 | eleqtrrd 2843 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ dom ∧ ) | 
| 15 | 10, 14 | jca 511 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) | 
| 16 | latpos 18484 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
| 17 | 3, 5 | meetcom 18450 | . . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) | 
| 18 | 16, 17 | syl3anl1 1413 | . 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 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 〈cop 4631 × cxp 5682 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Posetcpo 18354 joincjn 18358 meetcmee 18359 Latclat 18477 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-glb 18393 df-meet 18395 df-lat 18478 | 
| This theorem is referenced by: latleeqm2 18514 latmlem2 18516 latmlej21 18526 latmlej22 18527 mod2ile 18540 olm12 39230 latm12 39232 latm32 39233 latmrot 39234 olm02 39239 omllaw2N 39246 cmtcomlemN 39250 cmtbr3N 39256 omlfh1N 39260 omlmod1i2N 39262 omlspjN 39263 cvlcvrp 39342 intnatN 39410 cvrexch 39423 cvrat4 39446 2atjm 39448 1cvrat 39479 2at0mat0 39528 dalem4 39668 dalem56 39731 atmod2i1 39864 atmod2i2 39865 llnmod2i2 39866 atmod3i1 39867 atmod3i2 39868 llnexchb2lem 39871 dalawlem3 39876 dalawlem4 39877 dalawlem6 39879 dalawlem9 39882 dalawlem11 39884 dalawlem12 39885 dalawlem15 39888 lhpmcvr 40026 4atexlemc 40072 cdleme20zN 40304 cdleme20d 40315 cdleme20l 40325 cdleme20m 40326 cdlemg12 40653 cdlemg17 40680 cdlemg19 40687 cdlemg44a 40734 dihmeetlem17N 41326 dihmeetlem20N 41329 dihmeetALTN 41330 | 
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