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| Description: The join of a lattice commutes. (chjcom 31526 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 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 | latjcom.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
| 4 | latjcom.j | . . . . . . 7 ⊢ ∨ = (join‘𝐾) | |
| 5 | eqid 2736 | . . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
| 6 | 3, 4, 5 | islat 18479 | . . . . . 6 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom (meet‘𝐾) = (𝐵 × 𝐵)))) | 
| 7 | simprl 770 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom (meet‘𝐾) = (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, 4 | joincom 18448 | . . 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-lub 18392 df-join 18394 df-lat 18478 | 
| This theorem is referenced by: latleeqj2 18498 latjlej2 18500 latnle 18519 latmlej12 18525 latj12 18530 latj32 18531 latj13 18532 latj31 18533 latj4rot 18536 mod2ile 18540 latdisdlem 18542 olj02 39228 omllaw4 39248 cmt2N 39252 cmtbr3N 39256 cvlexch2 39331 cvlexchb2 39333 cvlatexchb2 39337 cvlatexch2 39339 cvlatexch3 39340 cvlatcvr2 39344 cvlsupr2 39345 cvlsupr7 39350 cvlsupr8 39351 hlatjcom 39370 hlrelat5N 39404 cvrval5 39418 cvrexch 39423 cvratlem 39424 cvrat 39425 2atlt 39442 cvrat3 39445 cvrat4 39446 cvrat42 39447 4noncolr3 39456 1cvrat 39479 3atlem1 39486 4atlem4d 39605 4atlem12 39615 paddcom 39816 paddasslem2 39824 pmapjat2 39857 atmod2i1 39864 atmod2i2 39865 llnmod2i2 39866 atmod4i1 39869 atmod4i2 39870 dalawlem4 39877 dalawlem9 39882 dalawlem12 39885 lhpjat2 40024 lhple 40045 trljat1 40169 trljat2 40170 cdlemc1 40194 cdlemc6 40199 cdlemd1 40201 cdleme5 40243 cdleme9 40256 cdleme10 40257 cdleme19e 40310 trlcolem 40729 trljco2 40744 cdlemk7 40851 cdlemk7u 40873 cdlemkid1 40925 dih1 41289 dihjatc2N 41315 | 
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