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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 1132 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
5 | simp2 1133 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | simp3 1134 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
7 | eqid 2823 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | 1, 7, 3, 4, 5, 6 | latcl2 17660 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom (join‘𝐾) ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
9 | 8 | simprd 498 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
10 | 1, 2, 3, 4, 5, 6, 9 | lemeet1 17638 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 joincjn 17556 meetcmee 17557 Latclat 17657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-glb 17587 df-meet 17589 df-lat 17658 |
This theorem is referenced by: latleeqm1 17691 latmlem1 17693 latnlemlt 17696 latmidm 17698 latabs1 17699 latledi 17701 latmlej11 17702 oldmm1 36355 cmtbr3N 36392 cmtbr4N 36393 lecmtN 36394 cvrat4 36581 2llnmat 36662 llnmlplnN 36677 dalem3 36802 dalem27 36837 dalem54 36864 dalem55 36865 2lnat 36922 cdlema1N 36929 llnexchb2lem 37006 dalawlem1 37009 dalawlem6 37014 dalawlem11 37019 dalawlem12 37020 4atexlemunv 37204 4atexlemc 37207 4atexlemnclw 37208 4atexlemex2 37209 4atexlemcnd 37210 lautm 37232 trlval3 37325 cdlemeulpq 37358 cdleme3h 37373 cdleme4a 37377 cdleme9 37391 cdleme11g 37403 cdleme13 37410 cdleme16e 37420 cdlemednpq 37437 cdleme19b 37442 cdleme20e 37451 cdleme20j 37456 cdleme22cN 37480 cdleme22e 37482 cdleme22eALTN 37483 cdleme22g 37486 cdleme35b 37588 cdleme35f 37592 cdlemeg46vrg 37665 cdlemg11b 37780 cdlemg12f 37786 cdlemg19a 37821 cdlemg31a 37835 cdlemk12 37988 cdlemkole 37991 cdlemk12u 38010 cdlemk37 38052 dia2dimlem1 38202 dihopelvalcpre 38386 dihmeetlem1N 38428 dihglblem5apreN 38429 dihglblem2N 38432 dihmeetlem2N 38437 |
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