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Mirrors > Home > MPE Home > Th. List > latlej1 | Structured version Visualization version GIF version |
Description: A join's first argument is less than or equal to the join. (chub1 31437 analog.) (Contributed by NM, 17-Sep-2011.) |
Ref | Expression |
---|---|
latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
latlej.l | ⊢ ≤ = (le‘𝐾) |
latlej.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latlej1 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latlej.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latlej.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | latlej.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | simp1 1133 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
5 | simp2 1134 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | simp3 1135 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
7 | eqid 2726 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
8 | 1, 3, 7, 4, 5, 6 | latcl2 18456 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom (meet‘𝐾))) |
9 | 8 | simpld 493 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
10 | 1, 2, 3, 4, 5, 6, 9 | lejoin1 18404 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 〈cop 4629 class class class wbr 5145 dom cdm 5674 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 lecple 17268 joincjn 18331 meetcmee 18332 Latclat 18451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-lub 18366 df-join 18368 df-lat 18452 |
This theorem is referenced by: latjlej1 18473 latnlej 18476 latnlej2 18479 latjidm 18482 latnle 18493 latabs2 18496 latmlej11 18498 latjass 18503 mod1ile 18513 lubun 18535 oldmm1 38928 olj01 38936 omllaw5N 38958 cvlexchb1 39041 cvlsupr2 39054 cvlsupr7 39059 hlatlej1 39086 hlrelat5N 39113 2atjm 39157 2llnmj 39272 lplnexllnN 39276 2llnjaN 39278 2llnm2N 39280 4atlem3a 39309 2lplnja 39331 2lplnm2N 39333 2lplnmj 39334 dalemply 39366 dalemsly 39367 dalem10 39385 dalem13 39388 dalem21 39406 dalem55 39439 2llnma1b 39498 cdlema1N 39503 elpaddn0 39512 paddasslem12 39543 paddasslem13 39544 pmapjoin 39564 dalawlem2 39584 dalawlem7 39589 dalawlem11 39593 dalawlem12 39594 lhpmcvr3 39737 lhpmcvr5N 39739 lhpmcvr6N 39740 lautj 39805 trljat1 39878 cdlemc1 39903 cdlemc4 39906 cdleme1 39939 cdleme8 39962 cdleme11g 39977 cdleme22e 40056 cdleme22eALTN 40057 cdleme23b 40062 cdleme23c 40063 cdleme27N 40081 cdleme30a 40090 cdleme35fnpq 40161 cdleme35b 40162 cdleme35c 40163 cdleme42h 40194 cdleme42i 40195 cdleme48bw 40214 cdlemg2fv2 40312 cdlemg7fvbwN 40319 cdlemg8b 40340 cdlemg11b 40354 trlcolem 40438 trljco 40452 cdlemi1 40530 cdlemk48 40662 cdlemn2 40907 dihjustlem 40928 dihord1 40930 dihord5apre 40974 dihglbcpreN 41012 dihmeetlem3N 41017 dihmeetlem11N 41029 |
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