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Mirrors > Home > MPE Home > Th. List > latlej1 | Structured version Visualization version GIF version |
Description: A join's first argument is less than or equal to the join. (chub1 27750 analog.) (Contributed by NM, 17-Sep-2011.) |
Ref | Expression |
---|---|
latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
latlej.l | ⊢ ≤ = (le‘𝐾) |
latlej.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latlej1 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ (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 | lejoin1 16835 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ (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: latjlej1 16888 latnlej 16891 latnlej2 16894 latjidm 16897 latnle 16908 latabs2 16911 latmlej11 16913 latjass 16918 mod1ile 16928 lubun 16946 oldmm1 33522 olj01 33530 omllaw5N 33552 cvlexchb1 33635 cvlsupr2 33648 cvlsupr7 33653 hlatlej1 33679 hlrelat5N 33705 2atjm 33749 2llnmj 33864 lplnexllnN 33868 2llnjaN 33870 2llnm2N 33872 4atlem3a 33901 2lplnja 33923 2lplnm2N 33925 2lplnmj 33926 dalemply 33958 dalemsly 33959 dalem10 33977 dalem13 33980 dalem21 33998 dalem55 34031 2llnma1b 34090 cdlema1N 34095 elpaddn0 34104 paddasslem12 34135 paddasslem13 34136 pmapjoin 34156 dalawlem2 34176 dalawlem7 34181 dalawlem11 34185 dalawlem12 34186 lhpmcvr3 34329 lhpmcvr5N 34331 lhpmcvr6N 34332 lautj 34397 trljat1 34471 cdlemc1 34496 cdlemc4 34499 cdleme1 34532 cdleme8 34555 cdleme11g 34570 cdleme22e 34650 cdleme22eALTN 34651 cdleme23b 34656 cdleme23c 34657 cdleme27N 34675 cdleme30a 34684 cdleme35fnpq 34755 cdleme35b 34756 cdleme35c 34757 cdleme42h 34788 cdleme42i 34789 cdleme48bw 34808 cdlemg2fv2 34906 cdlemg7fvbwN 34913 cdlemg8b 34934 cdlemg11b 34948 trlcolem 35032 trljco 35046 cdlemi1 35124 cdlemk48 35256 cdlemn2 35502 dihjustlem 35523 dihord1 35525 dihord5apre 35569 dihglbcpreN 35607 dihmeetlem3N 35612 dihmeetlem11N 35624 |
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