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Mirrors > Home > MPE Home > Th. List > latlej2 | Structured version Visualization version GIF version |
Description: A join's second argument is less than or equal to the join. (chub2 27751 analog.) (Contributed by NM, 17-Sep-2011.) |
Ref | Expression |
---|---|
latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
latlej.l | ⊢ ≤ = (le‘𝐾) |
latlej.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latlej2 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latlej.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latlej.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | latlej.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | simp1 1054 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
5 | simp2 1055 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | simp3 1056 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
7 | eqid 2610 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
8 | 1, 3, 7, 4, 5, 6 | latcl2 16871 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom (meet‘𝐾))) |
9 | 8 | simpld 474 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
10 | 1, 2, 3, 4, 5, 6, 9 | lejoin2 16836 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 Latclat 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-lub 16797 df-join 16799 df-lat 16869 |
This theorem is referenced by: latleeqj1 16886 latjlej1 16888 latnlej 16891 latnlej2 16894 latjass 16918 lubun 16946 oldmm1 33522 cmtcomlemN 33553 cmtbr4N 33560 cvlexchb1 33635 cvlatexch1 33641 cvrval5 33719 2llnjaN 33870 4atlem3b 33902 2lplnja 33923 dalem5 33971 dalem17 33984 dalem39 34015 dalem43 34019 elpaddn0 34104 pmapjoin 34156 dalawlem2 34176 dalawlem11 34185 dalawlem12 34186 lautj 34397 trljat2 34472 cdleme0cq 34520 cdleme1 34532 cdleme3 34542 cdleme5 34545 cdleme7ga 34553 cdleme10 34559 cdleme15b 34580 cdleme16b 34584 cdleme20k 34625 cdleme22e 34650 cdleme22eALTN 34651 cdleme23c 34657 cdleme28a 34676 cdleme32e 34751 cdleme35a 34754 cdlemg4c 34918 cdlemg6c 34926 trlcolem 35032 cdlemi1 35124 dia2dimlem2 35372 cdlemm10N 35425 dihord2pre2 35533 dihord5apre 35569 dihjatc1 35618 |
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