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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 1137 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
| 5 | simp2 1138 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
| 6 | simp3 1139 | . 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 18481 | . . 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 18443 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 meetcmee 18358 Latclat 18476 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-glb 18392 df-meet 18394 df-lat 18477 |
| This theorem is referenced by: latleeqm1 18512 latmlem1 18514 latnlemlt 18517 latmidm 18519 latabs1 18520 latledi 18522 latmlej11 18523 oldmm1 39218 cmtbr3N 39255 cmtbr4N 39256 lecmtN 39257 cvrat4 39445 2llnmat 39526 llnmlplnN 39541 dalem3 39666 dalem27 39701 dalem54 39728 dalem55 39729 2lnat 39786 cdlema1N 39793 llnexchb2lem 39870 dalawlem1 39873 dalawlem6 39878 dalawlem11 39883 dalawlem12 39884 4atexlemunv 40068 4atexlemc 40071 4atexlemnclw 40072 4atexlemex2 40073 4atexlemcnd 40074 lautm 40096 trlval3 40189 cdlemeulpq 40222 cdleme3h 40237 cdleme4a 40241 cdleme9 40255 cdleme11g 40267 cdleme13 40274 cdleme16e 40284 cdlemednpq 40301 cdleme19b 40306 cdleme20e 40315 cdleme20j 40320 cdleme22cN 40344 cdleme22e 40346 cdleme22eALTN 40347 cdleme22g 40350 cdleme35b 40452 cdleme35f 40456 cdlemeg46vrg 40529 cdlemg11b 40644 cdlemg12f 40650 cdlemg19a 40685 cdlemg31a 40699 cdlemk12 40852 cdlemkole 40855 cdlemk12u 40874 cdlemk37 40916 dia2dimlem1 41066 dihopelvalcpre 41250 dihmeetlem1N 41292 dihglblem5apreN 41293 dihglblem2N 41296 dihmeetlem2N 41301 |
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