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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 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 | lemeet2 18444 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌) |
| 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: latmlem1 18514 latledi 18522 mod1ile 18538 oldmm1 39218 olm01 39237 cmtcomlemN 39249 cmtbr4N 39256 meetat 39297 cvrexchlem 39421 cvrat4 39445 2llnmj 39562 2lplnmj 39624 dalem25 39700 dalem54 39728 dalem57 39731 cdlema1N 39793 cdlemb 39796 llnexchb2lem 39870 llnexch2N 39872 dalawlem1 39873 dalawlem3 39875 pl42lem1N 39981 lhpelim 40039 lhpat3 40048 4atexlemunv 40068 4atexlemtlw 40069 4atexlemnclw 40072 4atexlemex2 40073 lautm 40096 trlle 40186 cdlemc2 40194 cdlemc5 40197 cdlemd2 40201 cdleme0b 40214 cdleme0c 40215 cdleme0fN 40220 cdleme01N 40223 cdleme0ex1N 40225 cdleme2 40230 cdleme3b 40231 cdleme3c 40232 cdleme3g 40236 cdleme3h 40237 cdleme7aa 40244 cdleme7c 40247 cdleme7d 40248 cdleme7e 40249 cdleme7ga 40250 cdleme11fN 40266 cdleme11k 40270 cdleme15d 40279 cdleme16f 40285 cdlemednpq 40301 cdleme19c 40307 cdleme20aN 40311 cdleme20c 40313 cdleme20j 40320 cdleme21c 40329 cdleme21ct 40331 cdleme22cN 40344 cdleme22f 40348 cdleme23a 40351 cdleme28a 40372 cdleme35d 40454 cdleme35f 40456 cdlemeg46frv 40527 cdlemeg46rgv 40530 cdlemeg46req 40531 cdlemg2fv2 40602 cdlemg2m 40606 cdlemg4 40619 cdlemg10bALTN 40638 cdlemg31b 40700 trlcolem 40728 cdlemk14 40856 dia2dimlem1 41066 docaclN 41126 doca2N 41128 djajN 41139 dihjustlem 41218 dihord1 41220 dihord2a 41221 dihord2b 41222 dihord2cN 41223 dihord11b 41224 dihord11c 41226 dihord2pre 41227 dihlsscpre 41236 dihvalcq2 41249 dihopelvalcpre 41250 dihord6apre 41258 dihord5b 41261 dihord5apre 41264 dihmeetlem1N 41292 dihglblem5apreN 41293 dihglblem3N 41297 dihmeetbclemN 41306 dihmeetlem4preN 41308 dihmeetlem7N 41312 dihmeetlem9N 41317 dihjatcclem4 41423 |
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