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Mirrors > Home > MPE Home > Th. List > latlem12 | Structured version Visualization version GIF version |
Description: An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
latmle.b | ⊢ 𝐵 = (Base‘𝐾) |
latmle.l | ⊢ ≤ = (le‘𝐾) |
latmle.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latlem12 | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑍) ↔ 𝑋 ≤ (𝑌 ∧ 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latmle.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latmle.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | latmle.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
4 | latpos 18433 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
5 | 4 | adantr 479 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Poset) |
6 | simpr2 1192 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
7 | simpr3 1193 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
8 | simpr1 1191 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
9 | eqid 2725 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
10 | simpl 481 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
11 | 1, 9, 3, 10, 6, 7 | latcl2 18431 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (〈𝑌, 𝑍〉 ∈ dom (join‘𝐾) ∧ 〈𝑌, 𝑍〉 ∈ dom ∧ )) |
12 | 11 | simprd 494 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 〈𝑌, 𝑍〉 ∈ dom ∧ ) |
13 | 1, 2, 3, 5, 6, 7, 8, 12 | meetle 18395 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑍) ↔ 𝑋 ≤ (𝑌 ∧ 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 〈cop 4636 class class class wbr 5149 dom cdm 5678 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 lecple 17243 Posetcpo 18302 joincjn 18306 meetcmee 18307 Latclat 18426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-poset 18308 df-glb 18342 df-meet 18344 df-lat 18427 |
This theorem is referenced by: latleeqm1 18462 latmlem1 18464 latmidm 18469 latledi 18472 mod1ile 18488 oldmm1 38816 olm01 38835 cmtbr4N 38854 atnle 38916 atlatmstc 38918 hlrelat2 39003 cvrval5 39015 cvrexchlem 39019 2atjm 39045 atbtwn 39046 ps-2b 39082 2atm 39127 2llnm4 39170 2llnmeqat 39171 dalemcea 39260 dalem21 39294 dalem54 39326 dalem55 39327 dalem57 39329 2atm2atN 39385 2llnma1b 39386 cdlemblem 39393 dalawlem2 39472 dalawlem3 39473 dalawlem6 39476 dalawlem11 39481 dalawlem12 39482 lhpocnle 39616 lhpmcvr4N 39626 lhpat3 39646 4atexlemcnd 39672 lautm 39694 trlval3 39787 cdlemc5 39795 cdleme3 39837 cdleme7ga 39848 cdleme7 39849 cdleme11k 39868 cdleme16e 39882 cdleme16f 39883 cdlemednpq 39899 cdleme22aa 39939 cdleme22b 39941 cdleme22cN 39942 cdleme23c 39951 cdlemeg46req 40129 cdlemf2 40162 cdlemg10c 40239 cdlemg12f 40248 cdlemg17dALTN 40264 cdlemg19a 40283 cdlemg27b 40296 cdlemi 40420 cdlemk15 40455 cdlemk50 40552 dia2dimlem1 40664 dihopelvalcpre 40848 dihord5b 40859 dihmeetlem1N 40890 dihglblem5apreN 40891 dihglblem2N 40894 dihmeetlem3N 40905 |
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