Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1054 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
5 | simp2 1055 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | simp3 1056 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
7 | eqid 2610 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | 1, 7, 3, 4, 5, 6 | latcl2 16871 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom (join‘𝐾) ∧ 〈𝑋, 𝑌〉 ∈ dom ∧ )) |
9 | 8 | simprd 478 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
10 | 1, 2, 3, 4, 5, 6, 9 | lemeet2 16850 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 Latclat 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-glb 16798 df-meet 16800 df-lat 16869 |
This theorem is referenced by: latmlem1 16904 latledi 16912 mod1ile 16928 oldmm1 33522 olm01 33541 cmtcomlemN 33553 cmtbr4N 33560 meetat 33601 cvrexchlem 33723 cvrat4 33747 2llnmj 33864 2lplnmj 33926 dalem25 34002 dalem54 34030 dalem57 34033 cdlema1N 34095 cdlemb 34098 llnexchb2lem 34172 llnexch2N 34174 dalawlem1 34175 dalawlem3 34177 pl42lem1N 34283 lhpelim 34341 lhpat3 34350 4atexlemunv 34370 4atexlemtlw 34371 4atexlemnclw 34374 4atexlemex2 34375 lautm 34398 trlle 34489 cdlemc2 34497 cdlemc5 34500 cdlemd2 34504 cdleme0b 34517 cdleme0c 34518 cdleme0fN 34523 cdleme01N 34526 cdleme0ex1N 34528 cdleme2 34533 cdleme3b 34534 cdleme3c 34535 cdleme3g 34539 cdleme3h 34540 cdleme7aa 34547 cdleme7c 34550 cdleme7d 34551 cdleme7e 34552 cdleme7ga 34553 cdleme11fN 34569 cdleme11k 34573 cdleme15d 34582 cdleme16f 34588 cdlemednpq 34604 cdleme19c 34611 cdleme20aN 34615 cdleme20c 34617 cdleme20j 34624 cdleme21c 34633 cdleme21ct 34635 cdleme22cN 34648 cdleme22f 34652 cdleme23a 34655 cdleme28a 34676 cdleme35d 34758 cdleme35f 34760 cdlemeg46frv 34831 cdlemeg46rgv 34834 cdlemeg46req 34835 cdlemg2fv2 34906 cdlemg2m 34910 cdlemg4 34923 cdlemg10bALTN 34942 cdlemg31b 35004 trlcolem 35032 cdlemk14 35160 dia2dimlem1 35371 docaclN 35431 doca2N 35433 djajN 35444 dihjustlem 35523 dihord1 35525 dihord2a 35526 dihord2b 35527 dihord2cN 35528 dihord11b 35529 dihord11c 35531 dihord2pre 35532 dihlsscpre 35541 dihvalcq2 35554 dihopelvalcpre 35555 dihord6apre 35563 dihord5b 35566 dihord5apre 35569 dihmeetlem1N 35597 dihglblem5apreN 35598 dihglblem3N 35602 dihmeetbclemN 35611 dihmeetlem4preN 35613 dihmeetlem7N 35617 dihmeetlem9N 35622 dihjatcclem4 35728 |
Copyright terms: Public domain | W3C validator |