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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 17652 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
5 | 4 | adantr 484 | . 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 2798 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
10 | simpl 486 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
11 | 1, 9, 3, 10, 6, 7 | latcl2 17650 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (〈𝑌, 𝑍〉 ∈ dom (join‘𝐾) ∧ 〈𝑌, 𝑍〉 ∈ dom ∧ )) |
12 | 11 | simprd 499 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 〈𝑌, 𝑍〉 ∈ dom ∧ ) |
13 | 1, 2, 3, 5, 6, 7, 8, 12 | meetle 17630 | 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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 Posetcpo 17542 joincjn 17546 meetcmee 17547 Latclat 17647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 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 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-poset 17548 df-glb 17577 df-meet 17579 df-lat 17648 |
This theorem is referenced by: latleeqm1 17681 latmlem1 17683 latmidm 17688 latledi 17691 mod1ile 17707 oldmm1 36513 olm01 36532 cmtbr4N 36551 atnle 36613 atlatmstc 36615 hlrelat2 36699 cvrval5 36711 cvrexchlem 36715 2atjm 36741 atbtwn 36742 ps-2b 36778 2atm 36823 2llnm4 36866 2llnmeqat 36867 dalemcea 36956 dalem21 36990 dalem54 37022 dalem55 37023 dalem57 37025 2atm2atN 37081 2llnma1b 37082 cdlemblem 37089 dalawlem2 37168 dalawlem3 37169 dalawlem6 37172 dalawlem11 37177 dalawlem12 37178 lhpocnle 37312 lhpmcvr4N 37322 lhpat3 37342 4atexlemcnd 37368 lautm 37390 trlval3 37483 cdlemc5 37491 cdleme3 37533 cdleme7ga 37544 cdleme7 37545 cdleme11k 37564 cdleme16e 37578 cdleme16f 37579 cdlemednpq 37595 cdleme22aa 37635 cdleme22b 37637 cdleme22cN 37638 cdleme23c 37647 cdlemeg46req 37825 cdlemf2 37858 cdlemg10c 37935 cdlemg12f 37944 cdlemg17dALTN 37960 cdlemg19a 37979 cdlemg27b 37992 cdlemi 38116 cdlemk15 38151 cdlemk50 38248 dia2dimlem1 38360 dihopelvalcpre 38544 dihord5b 38555 dihmeetlem1N 38586 dihglblem5apreN 38587 dihglblem2N 38590 dihmeetlem3N 38601 |
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