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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 28695 analog.) (Contributed by NM, 16-Sep-2011.) |
Ref | Expression |
---|---|
latjcom.b | ⊢ 𝐵 = (Base‘𝐾) |
latjcom.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latjcom | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5305 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
2 | 1 | 3adant1 1125 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
3 | latjcom.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
4 | latjcom.j | . . . . . . 7 ⊢ ∨ = (join‘𝐾) | |
5 | eqid 2760 | . . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | 3, 4, 5 | islat 17268 | . . . . . 6 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom (meet‘𝐾) = (𝐵 × 𝐵)))) |
7 | simprl 811 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom (meet‘𝐾) = (𝐵 × 𝐵))) → dom ∨ = (𝐵 × 𝐵)) | |
8 | 6, 7 | sylbi 207 | . . . . 5 ⊢ (𝐾 ∈ Lat → dom ∨ = (𝐵 × 𝐵)) |
9 | 8 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → dom ∨ = (𝐵 × 𝐵)) |
10 | 2, 9 | eleqtrrd 2842 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
11 | opelxpi 5305 | . . . . . 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 1125 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
14 | 13, 9 | eleqtrrd 2842 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ dom ∨ ) |
15 | 10, 14 | jca 555 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑌, 𝑋〉 ∈ dom ∨ )) |
16 | latpos 17271 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
17 | 3, 4 | joincom 17251 | . . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑌, 𝑋〉 ∈ dom ∨ )) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋)) |
18 | 16, 17 | syl3anl1 1521 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑌, 𝑋〉 ∈ dom ∨ )) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋)) |
19 | 15, 18 | mpdan 705 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 〈cop 4327 × cxp 5264 dom cdm 5266 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 Posetcpo 17161 joincjn 17165 meetcmee 17166 Latclat 17266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-lub 17195 df-join 17197 df-lat 17267 |
This theorem is referenced by: latleeqj2 17285 latjlej2 17287 latnle 17306 latmlej12 17312 latj12 17317 latj32 17318 latj13 17319 latj31 17320 latj4rot 17323 mod2ile 17327 latdisdlem 17410 olj02 35034 omllaw4 35054 cmt2N 35058 cmtbr3N 35062 cvlexch2 35137 cvlexchb2 35139 cvlatexchb2 35143 cvlatexch2 35145 cvlatexch3 35146 cvlatcvr2 35150 cvlsupr2 35151 cvlsupr7 35156 cvlsupr8 35157 hlatjcom 35175 hlrelat5N 35208 cvrval5 35222 cvrexch 35227 cvratlem 35228 cvrat 35229 2atlt 35246 cvrat3 35249 cvrat4 35250 cvrat42 35251 4noncolr3 35260 1cvrat 35283 3atlem1 35290 4atlem4d 35409 4atlem12 35419 paddcom 35620 paddasslem2 35628 pmapjat2 35661 atmod2i1 35668 atmod2i2 35669 llnmod2i2 35670 atmod4i1 35673 atmod4i2 35674 dalawlem4 35681 dalawlem9 35686 dalawlem12 35689 lhpjat2 35828 lhple 35849 trljat1 35974 trljat2 35975 cdlemc1 35999 cdlemc6 36004 cdlemd1 36006 cdleme5 36048 cdleme9 36061 cdleme10 36062 cdleme19e 36115 trlcolem 36534 trljco2 36549 cdlemk7 36656 cdlemk7u 36678 cdlemkid1 36730 dih1 37095 dihjatc2N 37121 |
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