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Mirrors > Home > MPE Home > Th. List > latmle1 | Structured version Visualization version GIF version |
Description: A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
latmle.b | ⊢ 𝐵 = (Base‘𝐾) |
latmle.l | ⊢ ≤ = (le‘𝐾) |
latmle.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latmle1 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋) |
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 | lemeet1 17904 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋) |
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: latleeqm1 17973 latmlem1 17975 latnlemlt 17978 latmidm 17980 latabs1 17981 latledi 17983 latmlej11 17984 oldmm1 36968 cmtbr3N 37005 cmtbr4N 37006 lecmtN 37007 cvrat4 37194 2llnmat 37275 llnmlplnN 37290 dalem3 37415 dalem27 37450 dalem54 37477 dalem55 37478 2lnat 37535 cdlema1N 37542 llnexchb2lem 37619 dalawlem1 37622 dalawlem6 37627 dalawlem11 37632 dalawlem12 37633 4atexlemunv 37817 4atexlemc 37820 4atexlemnclw 37821 4atexlemex2 37822 4atexlemcnd 37823 lautm 37845 trlval3 37938 cdlemeulpq 37971 cdleme3h 37986 cdleme4a 37990 cdleme9 38004 cdleme11g 38016 cdleme13 38023 cdleme16e 38033 cdlemednpq 38050 cdleme19b 38055 cdleme20e 38064 cdleme20j 38069 cdleme22cN 38093 cdleme22e 38095 cdleme22eALTN 38096 cdleme22g 38099 cdleme35b 38201 cdleme35f 38205 cdlemeg46vrg 38278 cdlemg11b 38393 cdlemg12f 38399 cdlemg19a 38434 cdlemg31a 38448 cdlemk12 38601 cdlemkole 38604 cdlemk12u 38623 cdlemk37 38665 dia2dimlem1 38815 dihopelvalcpre 38999 dihmeetlem1N 39041 dihglblem5apreN 39042 dihglblem2N 39045 dihmeetlem2N 39050 |
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