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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 1128 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
5 | simp2 1129 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | simp3 1130 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
7 | eqid 2818 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | 1, 7, 3, 4, 5, 6 | latcl2 17646 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom (join‘𝐾) ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
9 | 8 | simprd 496 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
10 | 1, 2, 3, 4, 5, 6, 9 | lemeet1 17624 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 joincjn 17542 meetcmee 17543 Latclat 17643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-glb 17573 df-meet 17575 df-lat 17644 |
This theorem is referenced by: latleeqm1 17677 latmlem1 17679 latnlemlt 17682 latmidm 17684 latabs1 17685 latledi 17687 latmlej11 17688 oldmm1 36233 cmtbr3N 36270 cmtbr4N 36271 lecmtN 36272 cvrat4 36459 2llnmat 36540 llnmlplnN 36555 dalem3 36680 dalem27 36715 dalem54 36742 dalem55 36743 2lnat 36800 cdlema1N 36807 llnexchb2lem 36884 dalawlem1 36887 dalawlem6 36892 dalawlem11 36897 dalawlem12 36898 4atexlemunv 37082 4atexlemc 37085 4atexlemnclw 37086 4atexlemex2 37087 4atexlemcnd 37088 lautm 37110 trlval3 37203 cdlemeulpq 37236 cdleme3h 37251 cdleme4a 37255 cdleme9 37269 cdleme11g 37281 cdleme13 37288 cdleme16e 37298 cdlemednpq 37315 cdleme19b 37320 cdleme20e 37329 cdleme20j 37334 cdleme22cN 37358 cdleme22e 37360 cdleme22eALTN 37361 cdleme22g 37364 cdleme35b 37466 cdleme35f 37470 cdlemeg46vrg 37543 cdlemg11b 37658 cdlemg12f 37664 cdlemg19a 37699 cdlemg31a 37713 cdlemk12 37866 cdlemkole 37869 cdlemk12u 37888 cdlemk37 37930 dia2dimlem1 38080 dihopelvalcpre 38264 dihmeetlem1N 38306 dihglblem5apreN 38307 dihglblem2N 38310 dihmeetlem2N 38315 |
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