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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 5719 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
2 | 1 | 3adant1 1127 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
3 | latmcom.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
4 | eqid 2726 | . . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | latmcom.m | . . . . . . 7 ⊢ ∧ = (meet‘𝐾) | |
6 | 3, 4, 5 | islat 18458 | . . . . . 6 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵)))) |
7 | simprr 771 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (dom (join‘𝐾) = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))) → dom ∧ = (𝐵 × 𝐵)) | |
8 | 6, 7 | sylbi 216 | . . . . 5 ⊢ (𝐾 ∈ Lat → dom ∧ = (𝐵 × 𝐵)) |
9 | 8 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → dom ∧ = (𝐵 × 𝐵)) |
10 | 2, 9 | eleqtrrd 2829 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
11 | opelxpi 5719 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) | |
12 | 11 | ancoms 457 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
13 | 12 | 3adant1 1127 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
14 | 13, 9 | eleqtrrd 2829 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ dom ∧ ) |
15 | 10, 14 | jca 510 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) |
16 | latpos 18463 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
17 | 3, 5 | meetcom 18429 | . . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
18 | 16, 17 | syl3anl1 1409 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∧ ∧ 〈𝑌, 𝑋〉 ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
19 | 15, 18 | mpdan 685 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 〈cop 4639 × cxp 5680 dom cdm 5682 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 Posetcpo 18332 joincjn 18336 meetcmee 18337 Latclat 18456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-glb 18372 df-meet 18374 df-lat 18457 |
This theorem is referenced by: latleeqm2 18493 latmlem2 18495 latmlej21 18505 latmlej22 18506 mod2ile 18519 olm12 38926 latm12 38928 latm32 38929 latmrot 38930 olm02 38935 omllaw2N 38942 cmtcomlemN 38946 cmtbr3N 38952 omlfh1N 38956 omlmod1i2N 38958 omlspjN 38959 cvlcvrp 39038 intnatN 39106 cvrexch 39119 cvrat4 39142 2atjm 39144 1cvrat 39175 2at0mat0 39224 dalem4 39364 dalem56 39427 atmod2i1 39560 atmod2i2 39561 llnmod2i2 39562 atmod3i1 39563 atmod3i2 39564 llnexchb2lem 39567 dalawlem3 39572 dalawlem4 39573 dalawlem6 39575 dalawlem9 39578 dalawlem11 39580 dalawlem12 39581 dalawlem15 39584 lhpmcvr 39722 4atexlemc 39768 cdleme20zN 40000 cdleme20d 40011 cdleme20l 40021 cdleme20m 40022 cdlemg12 40349 cdlemg17 40376 cdlemg19 40383 cdlemg44a 40430 dihmeetlem17N 41022 dihmeetlem20N 41025 dihmeetALTN 41026 |
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