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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 29433 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 5556 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
2 | 1 | 3adant1 1131 | . . . 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 2738 | . . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | 3, 4, 5 | islat 17766 | . . . . . 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 220 | . . . . 5 ⊢ (𝐾 ∈ Lat → dom ∨ = (𝐵 × 𝐵)) |
9 | 8 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → dom ∨ = (𝐵 × 𝐵)) |
10 | 2, 9 | eleqtrrd 2836 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
11 | opelxpi 5556 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) | |
12 | 11 | ancoms 462 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
13 | 12 | 3adant1 1131 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ (𝐵 × 𝐵)) |
14 | 13, 9 | eleqtrrd 2836 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑌, 𝑋〉 ∈ dom ∨ ) |
15 | 10, 14 | jca 515 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑌, 𝑋〉 ∈ dom ∨ )) |
16 | latpos 17769 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
17 | 3, 4 | joincom 17749 | . . 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 〈cop 4519 × cxp 5517 dom cdm 5519 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 Posetcpo 17659 joincjn 17663 meetcmee 17664 Latclat 17764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-lub 17693 df-join 17695 df-lat 17765 |
This theorem is referenced by: latleeqj2 17783 latjlej2 17785 latnle 17804 latmlej12 17810 latj12 17815 latj32 17816 latj13 17817 latj31 17818 latj4rot 17821 mod2ile 17825 latdisdlem 17908 olj02 36852 omllaw4 36872 cmt2N 36876 cmtbr3N 36880 cvlexch2 36955 cvlexchb2 36957 cvlatexchb2 36961 cvlatexch2 36963 cvlatexch3 36964 cvlatcvr2 36968 cvlsupr2 36969 cvlsupr7 36974 cvlsupr8 36975 hlatjcom 36994 hlrelat5N 37027 cvrval5 37041 cvrexch 37046 cvratlem 37047 cvrat 37048 2atlt 37065 cvrat3 37068 cvrat4 37069 cvrat42 37070 4noncolr3 37079 1cvrat 37102 3atlem1 37109 4atlem4d 37228 4atlem12 37238 paddcom 37439 paddasslem2 37447 pmapjat2 37480 atmod2i1 37487 atmod2i2 37488 llnmod2i2 37489 atmod4i1 37492 atmod4i2 37493 dalawlem4 37500 dalawlem9 37505 dalawlem12 37508 lhpjat2 37647 lhple 37668 trljat1 37792 trljat2 37793 cdlemc1 37817 cdlemc6 37822 cdlemd1 37824 cdleme5 37866 cdleme9 37879 cdleme10 37880 cdleme19e 37933 trlcolem 38352 trljco2 38367 cdlemk7 38474 cdlemk7u 38496 cdlemkid1 38548 dih1 38912 dihjatc2N 38938 |
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