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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 29214 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 17650 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
5 | 4 | adantr 481 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Poset) |
6 | simpr1 1186 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
7 | simpr2 1187 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
8 | simpr3 1188 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
9 | eqid 2821 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
10 | simpl 483 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
11 | 1, 3, 9, 10, 6, 7 | latcl2 17648 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom (meet‘𝐾))) |
12 | 11 | simpld 495 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
13 | 1, 2, 3, 5, 6, 7, 8, 12 | joinle 17614 | 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 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 〈cop 4565 class class class wbr 5058 dom cdm 5549 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 lecple 16562 Posetcpo 17540 joincjn 17544 meetcmee 17545 Latclat 17645 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-poset 17546 df-lub 17574 df-join 17576 df-lat 17646 |
This theorem is referenced by: latleeqj1 17663 latjlej1 17665 latjidm 17674 latledi 17689 latjass 17695 mod1ile 17705 lubun 17723 oldmm1 36235 olj01 36243 cvlexchb1 36348 cvlcvr1 36357 hlrelat 36420 hlrelat2 36421 exatleN 36422 hlrelat3 36430 cvrexchlem 36437 cvratlem 36439 cvrat 36440 atlelt 36456 ps-1 36495 hlatexch3N 36498 hlatexch4 36499 3atlem1 36501 3atlem2 36502 lplnexllnN 36582 2llnjaN 36584 4atlem3 36614 4atlem10 36624 4atlem11b 36626 4atlem11 36627 4atlem12b 36629 4atlem12 36630 2lplnja 36637 dalem1 36677 dalem3 36682 dalem8 36688 dalem16 36697 dalem17 36698 dalem21 36712 dalem25 36716 dalem39 36729 dalem54 36744 dalem60 36750 linepsubN 36770 pmapsub 36786 lneq2at 36796 2llnma3r 36806 cdlema1N 36809 cdlemblem 36811 paddasslem5 36842 paddasslem12 36849 paddasslem13 36850 llnexchb2 36887 dalawlem3 36891 dalawlem5 36893 dalawlem8 36896 dalawlem11 36899 dalawlem12 36900 lhp2lt 37019 lhpexle2lem 37027 lhpexle3lem 37029 4atexlemtlw 37085 4atexlemnclw 37088 lautj 37111 cdlemd3 37218 cdleme3g 37252 cdleme3h 37253 cdleme7d 37264 cdleme11c 37279 cdleme15d 37295 cdleme17b 37305 cdleme19a 37321 cdleme20j 37336 cdleme21c 37345 cdleme22b 37359 cdleme22d 37361 cdleme28a 37388 cdleme35a 37466 cdleme35fnpq 37467 cdleme35b 37468 cdleme35f 37472 cdleme42c 37490 cdleme42i 37501 cdlemf1 37579 cdlemg4c 37630 cdlemg6c 37638 cdlemg8b 37646 cdlemg10 37659 cdlemg11b 37660 cdlemg13a 37669 cdlemg17a 37679 cdlemg18b 37697 cdlemg27a 37710 cdlemg33b0 37719 cdlemg35 37731 cdlemg42 37747 cdlemg46 37753 trljco 37758 tendopltp 37798 cdlemk3 37851 cdlemk10 37861 cdlemk1u 37877 cdlemk39 37934 dialss 38064 dia2dimlem1 38082 dia2dimlem10 38091 dia2dimlem12 38093 cdlemm10N 38136 djajN 38155 diblss 38188 cdlemn2 38213 dihord2pre2 38244 dib2dim 38261 dih2dimb 38262 dih2dimbALTN 38263 dihmeetlem6 38327 dihjatcclem1 38436 |
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