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Mirrors > Home > MPE Home > Th. List > latmle2 | Structured version Visualization version GIF version |
Description: A meet is less than or equal to its second argument. (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
latmle.b | ⊢ 𝐵 = (Base‘𝐾) |
latmle.l | ⊢ ≤ = (le‘𝐾) |
latmle.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latmle2 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latmle.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latmle.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | latmle.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
4 | simp1 1138 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
5 | simp2 1139 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | simp3 1140 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
7 | eqid 2737 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | 1, 7, 3, 4, 5, 6 | latcl2 17942 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom (join‘𝐾) ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
9 | 8 | simprd 499 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
10 | 1, 2, 3, 4, 5, 6, 9 | lemeet2 17905 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 〈cop 4547 class class class wbr 5053 dom cdm 5551 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 lecple 16809 joincjn 17818 meetcmee 17819 Latclat 17937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-glb 17853 df-meet 17855 df-lat 17938 |
This theorem is referenced by: latmlem1 17975 latledi 17983 mod1ile 17999 oldmm1 36968 olm01 36987 cmtcomlemN 36999 cmtbr4N 37006 meetat 37047 cvrexchlem 37170 cvrat4 37194 2llnmj 37311 2lplnmj 37373 dalem25 37449 dalem54 37477 dalem57 37480 cdlema1N 37542 cdlemb 37545 llnexchb2lem 37619 llnexch2N 37621 dalawlem1 37622 dalawlem3 37624 pl42lem1N 37730 lhpelim 37788 lhpat3 37797 4atexlemunv 37817 4atexlemtlw 37818 4atexlemnclw 37821 4atexlemex2 37822 lautm 37845 trlle 37935 cdlemc2 37943 cdlemc5 37946 cdlemd2 37950 cdleme0b 37963 cdleme0c 37964 cdleme0fN 37969 cdleme01N 37972 cdleme0ex1N 37974 cdleme2 37979 cdleme3b 37980 cdleme3c 37981 cdleme3g 37985 cdleme3h 37986 cdleme7aa 37993 cdleme7c 37996 cdleme7d 37997 cdleme7e 37998 cdleme7ga 37999 cdleme11fN 38015 cdleme11k 38019 cdleme15d 38028 cdleme16f 38034 cdlemednpq 38050 cdleme19c 38056 cdleme20aN 38060 cdleme20c 38062 cdleme20j 38069 cdleme21c 38078 cdleme21ct 38080 cdleme22cN 38093 cdleme22f 38097 cdleme23a 38100 cdleme28a 38121 cdleme35d 38203 cdleme35f 38205 cdlemeg46frv 38276 cdlemeg46rgv 38279 cdlemeg46req 38280 cdlemg2fv2 38351 cdlemg2m 38355 cdlemg4 38368 cdlemg10bALTN 38387 cdlemg31b 38449 trlcolem 38477 cdlemk14 38605 dia2dimlem1 38815 docaclN 38875 doca2N 38877 djajN 38888 dihjustlem 38967 dihord1 38969 dihord2a 38970 dihord2b 38971 dihord2cN 38972 dihord11b 38973 dihord11c 38975 dihord2pre 38976 dihlsscpre 38985 dihvalcq2 38998 dihopelvalcpre 38999 dihord6apre 39007 dihord5b 39010 dihord5apre 39013 dihmeetlem1N 39041 dihglblem5apreN 39042 dihglblem3N 39046 dihmeetbclemN 39055 dihmeetlem4preN 39057 dihmeetlem7N 39061 dihmeetlem9N 39066 dihjatcclem4 39172 |
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