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Mirrors > Home > MPE Home > Th. List > latjle12 | Structured version Visualization version GIF version |
Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 27752 analog.) (Contributed by NM, 17-Sep-2011.) |
Ref | Expression |
---|---|
latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
latlej.l | ⊢ ≤ = (le‘𝐾) |
latlej.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latjle12 | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latlej.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latlej.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | latlej.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | latpos 16873 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
5 | 4 | adantr 480 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Poset) |
6 | simpr1 1060 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
7 | simpr2 1061 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
8 | simpr3 1062 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
9 | eqid 2610 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
10 | simpl 472 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
11 | 1, 3, 9, 10, 6, 7 | latcl2 16871 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom (meet‘𝐾))) |
12 | 11 | simpld 474 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
13 | 1, 2, 3, 5, 6, 7, 8, 12 | joinle 16837 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ 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 Posetcpo 16763 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-poset 16769 df-lub 16797 df-join 16799 df-lat 16869 |
This theorem is referenced by: latleeqj1 16886 latjlej1 16888 latjidm 16897 latledi 16912 latjass 16918 mod1ile 16928 lubun 16946 oldmm1 33522 olj01 33530 cvlexchb1 33635 cvlcvr1 33644 hlrelat 33706 hlrelat2 33707 exatleN 33708 hlrelat3 33716 cvrexchlem 33723 cvratlem 33725 cvrat 33726 atlelt 33742 ps-1 33781 hlatexch3N 33784 hlatexch4 33785 3atlem1 33787 3atlem2 33788 lplnexllnN 33868 2llnjaN 33870 4atlem3 33900 4atlem10 33910 4atlem11b 33912 4atlem11 33913 4atlem12b 33915 4atlem12 33916 2lplnja 33923 dalem1 33963 dalem3 33968 dalem8 33974 dalem16 33983 dalem17 33984 dalem21 33998 dalem25 34002 dalem39 34015 dalem54 34030 dalem60 34036 linepsubN 34056 pmapsub 34072 lneq2at 34082 2llnma3r 34092 cdlema1N 34095 cdlemblem 34097 paddasslem5 34128 paddasslem12 34135 paddasslem13 34136 llnexchb2 34173 dalawlem3 34177 dalawlem5 34179 dalawlem8 34182 dalawlem11 34185 dalawlem12 34186 lhp2lt 34305 lhpexle2lem 34313 lhpexle3lem 34315 4atexlemtlw 34371 4atexlemnclw 34374 lautj 34397 cdlemd3 34505 cdleme3g 34539 cdleme3h 34540 cdleme7d 34551 cdleme11c 34566 cdleme15d 34582 cdleme17b 34592 cdleme19a 34609 cdleme20j 34624 cdleme21c 34633 cdleme22b 34647 cdleme22d 34649 cdleme28a 34676 cdleme35a 34754 cdleme35fnpq 34755 cdleme35b 34756 cdleme35f 34760 cdleme42c 34778 cdleme42i 34789 cdlemf1 34867 cdlemg4c 34918 cdlemg6c 34926 cdlemg8b 34934 cdlemg10 34947 cdlemg11b 34948 cdlemg13a 34957 cdlemg17a 34967 cdlemg18b 34985 cdlemg27a 34998 cdlemg33b0 35007 cdlemg35 35019 cdlemg42 35035 cdlemg46 35041 trljco 35046 tendopltp 35086 cdlemk3 35139 cdlemk10 35149 cdlemk1u 35165 cdlemk39 35222 dialss 35353 dia2dimlem1 35371 dia2dimlem10 35380 dia2dimlem12 35382 cdlemm10N 35425 djajN 35444 diblss 35477 cdlemn2 35502 dihord2pre2 35533 dib2dim 35550 dih2dimb 35551 dih2dimbALTN 35552 dihmeetlem6 35616 dihjatcclem1 35725 |
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