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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 31535 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 5726 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
2 | 1 | 3adant1 1129 | . . . 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 2735 | . . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | 3, 4, 5 | islat 18491 | . . . . . 6 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom (meet‘𝐾) = (𝐵 × 𝐵)))) |
7 | simprl 771 | . . . . . 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 1132 | . . . 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 5726 | . . . . . 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 1129 | . . . 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 511 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑌, 𝑋〉 ∈ dom ∨ )) |
16 | latpos 18496 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
17 | 3, 4 | joincom 18460 | . . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑌, 𝑋〉 ∈ dom ∨ )) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋)) |
18 | 16, 17 | syl3anl1 1411 | . 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 1537 ∈ wcel 2106 〈cop 4637 × cxp 5687 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Posetcpo 18365 joincjn 18369 meetcmee 18370 Latclat 18489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-lub 18404 df-join 18406 df-lat 18490 |
This theorem is referenced by: latleeqj2 18510 latjlej2 18512 latnle 18531 latmlej12 18537 latj12 18542 latj32 18543 latj13 18544 latj31 18545 latj4rot 18548 mod2ile 18552 latdisdlem 18554 olj02 39208 omllaw4 39228 cmt2N 39232 cmtbr3N 39236 cvlexch2 39311 cvlexchb2 39313 cvlatexchb2 39317 cvlatexch2 39319 cvlatexch3 39320 cvlatcvr2 39324 cvlsupr2 39325 cvlsupr7 39330 cvlsupr8 39331 hlatjcom 39350 hlrelat5N 39384 cvrval5 39398 cvrexch 39403 cvratlem 39404 cvrat 39405 2atlt 39422 cvrat3 39425 cvrat4 39426 cvrat42 39427 4noncolr3 39436 1cvrat 39459 3atlem1 39466 4atlem4d 39585 4atlem12 39595 paddcom 39796 paddasslem2 39804 pmapjat2 39837 atmod2i1 39844 atmod2i2 39845 llnmod2i2 39846 atmod4i1 39849 atmod4i2 39850 dalawlem4 39857 dalawlem9 39862 dalawlem12 39865 lhpjat2 40004 lhple 40025 trljat1 40149 trljat2 40150 cdlemc1 40174 cdlemc6 40179 cdlemd1 40181 cdleme5 40223 cdleme9 40236 cdleme10 40237 cdleme19e 40290 trlcolem 40709 trljco2 40724 cdlemk7 40831 cdlemk7u 40853 cdlemkid1 40905 dih1 41269 dihjatc2N 41295 |
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