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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 1135 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
5 | simp2 1136 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | simp3 1137 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
7 | eqid 2735 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | 1, 7, 3, 4, 5, 6 | latcl2 18494 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom (join‘𝐾) ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
9 | 8 | simprd 495 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
10 | 1, 2, 3, 4, 5, 6, 9 | lemeet1 18456 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 joincjn 18369 meetcmee 18370 Latclat 18489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-glb 18405 df-meet 18407 df-lat 18490 |
This theorem is referenced by: latleeqm1 18525 latmlem1 18527 latnlemlt 18530 latmidm 18532 latabs1 18533 latledi 18535 latmlej11 18536 oldmm1 39199 cmtbr3N 39236 cmtbr4N 39237 lecmtN 39238 cvrat4 39426 2llnmat 39507 llnmlplnN 39522 dalem3 39647 dalem27 39682 dalem54 39709 dalem55 39710 2lnat 39767 cdlema1N 39774 llnexchb2lem 39851 dalawlem1 39854 dalawlem6 39859 dalawlem11 39864 dalawlem12 39865 4atexlemunv 40049 4atexlemc 40052 4atexlemnclw 40053 4atexlemex2 40054 4atexlemcnd 40055 lautm 40077 trlval3 40170 cdlemeulpq 40203 cdleme3h 40218 cdleme4a 40222 cdleme9 40236 cdleme11g 40248 cdleme13 40255 cdleme16e 40265 cdlemednpq 40282 cdleme19b 40287 cdleme20e 40296 cdleme20j 40301 cdleme22cN 40325 cdleme22e 40327 cdleme22eALTN 40328 cdleme22g 40331 cdleme35b 40433 cdleme35f 40437 cdlemeg46vrg 40510 cdlemg11b 40625 cdlemg12f 40631 cdlemg19a 40666 cdlemg31a 40680 cdlemk12 40833 cdlemkole 40836 cdlemk12u 40855 cdlemk37 40897 dia2dimlem1 41047 dihopelvalcpre 41231 dihmeetlem1N 41273 dihglblem5apreN 41274 dihglblem2N 41277 dihmeetlem2N 41282 |
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