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| Mirrors > Home > MPE Home > Th. List > latjle12 | Structured version Visualization version GIF version | ||
| Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 31528 analog.) (Contributed by NM, 17-Sep-2011.) |
| Ref | Expression |
|---|---|
| latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
| latlej.l | ⊢ ≤ = (le‘𝐾) |
| latlej.j | ⊢ ∨ = (join‘𝐾) |
| Ref | Expression |
|---|---|
| latjle12 | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latlej.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | latlej.l | . 2 ⊢ ≤ = (le‘𝐾) | |
| 3 | latlej.j | . 2 ⊢ ∨ = (join‘𝐾) | |
| 4 | latpos 18483 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
| 5 | 4 | adantr 480 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Poset) |
| 6 | simpr1 1195 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
| 7 | simpr2 1196 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
| 8 | simpr3 1197 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
| 9 | eqid 2737 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
| 10 | simpl 482 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
| 11 | 1, 3, 9, 10, 6, 7 | latcl2 18481 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom (meet‘𝐾))) |
| 12 | 11 | simpld 494 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
| 13 | 1, 2, 3, 5, 6, 7, 8, 12 | joinle 18431 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ 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 Posetcpo 18353 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-poset 18359 df-lub 18391 df-join 18393 df-lat 18477 |
| This theorem is referenced by: latleeqj1 18496 latjlej1 18498 latjidm 18507 latledi 18522 latjass 18528 mod1ile 18538 lubun 18560 oldmm1 39218 olj01 39226 cvlexchb1 39331 cvlcvr1 39340 hlrelat 39404 hlrelat2 39405 exatleN 39406 hlrelat3 39414 cvrexchlem 39421 cvratlem 39423 cvrat 39424 atlelt 39440 ps-1 39479 hlatexch3N 39482 hlatexch4 39483 3atlem1 39485 3atlem2 39486 lplnexllnN 39566 2llnjaN 39568 4atlem3 39598 4atlem10 39608 4atlem11b 39610 4atlem11 39611 4atlem12b 39613 4atlem12 39614 2lplnja 39621 dalem1 39661 dalem3 39666 dalem8 39672 dalem16 39681 dalem17 39682 dalem21 39696 dalem25 39700 dalem39 39713 dalem54 39728 dalem60 39734 linepsubN 39754 pmapsub 39770 lneq2at 39780 2llnma3r 39790 cdlema1N 39793 cdlemblem 39795 paddasslem5 39826 paddasslem12 39833 paddasslem13 39834 llnexchb2 39871 dalawlem3 39875 dalawlem5 39877 dalawlem8 39880 dalawlem11 39883 dalawlem12 39884 lhp2lt 40003 lhpexle2lem 40011 lhpexle3lem 40013 4atexlemtlw 40069 4atexlemnclw 40072 lautj 40095 cdlemd3 40202 cdleme3g 40236 cdleme3h 40237 cdleme7d 40248 cdleme11c 40263 cdleme15d 40279 cdleme17b 40289 cdleme19a 40305 cdleme20j 40320 cdleme21c 40329 cdleme22b 40343 cdleme22d 40345 cdleme28a 40372 cdleme35a 40450 cdleme35fnpq 40451 cdleme35b 40452 cdleme35f 40456 cdleme42c 40474 cdleme42i 40485 cdlemf1 40563 cdlemg4c 40614 cdlemg6c 40622 cdlemg8b 40630 cdlemg10 40643 cdlemg11b 40644 cdlemg13a 40653 cdlemg17a 40663 cdlemg18b 40681 cdlemg27a 40694 cdlemg33b0 40703 cdlemg35 40715 cdlemg42 40731 cdlemg46 40737 trljco 40742 tendopltp 40782 cdlemk3 40835 cdlemk10 40845 cdlemk1u 40861 cdlemk39 40918 dialss 41048 dia2dimlem1 41066 dia2dimlem10 41075 dia2dimlem12 41077 cdlemm10N 41120 djajN 41139 diblss 41172 cdlemn2 41197 dihord2pre2 41228 dib2dim 41245 dih2dimb 41246 dih2dimbALTN 41247 dihmeetlem6 41311 dihjatcclem1 41420 |
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