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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 31439 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 18458 | . . 3 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
5 | 4 | adantr 479 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Poset) |
6 | simpr1 1191 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
7 | simpr2 1192 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
8 | simpr3 1193 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
9 | eqid 2726 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
10 | simpl 481 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
11 | 1, 3, 9, 10, 6, 7 | latcl2 18456 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom (meet‘𝐾))) |
12 | 11 | simpld 493 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
13 | 1, 2, 3, 5, 6, 7, 8, 12 | joinle 18406 | 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 205 ∧ wa 394 ∧ 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 Posetcpo 18327 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-poset 18333 df-lub 18366 df-join 18368 df-lat 18452 |
This theorem is referenced by: latleeqj1 18471 latjlej1 18473 latjidm 18482 latledi 18497 latjass 18503 mod1ile 18513 lubun 18535 oldmm1 38928 olj01 38936 cvlexchb1 39041 cvlcvr1 39050 hlrelat 39114 hlrelat2 39115 exatleN 39116 hlrelat3 39124 cvrexchlem 39131 cvratlem 39133 cvrat 39134 atlelt 39150 ps-1 39189 hlatexch3N 39192 hlatexch4 39193 3atlem1 39195 3atlem2 39196 lplnexllnN 39276 2llnjaN 39278 4atlem3 39308 4atlem10 39318 4atlem11b 39320 4atlem11 39321 4atlem12b 39323 4atlem12 39324 2lplnja 39331 dalem1 39371 dalem3 39376 dalem8 39382 dalem16 39391 dalem17 39392 dalem21 39406 dalem25 39410 dalem39 39423 dalem54 39438 dalem60 39444 linepsubN 39464 pmapsub 39480 lneq2at 39490 2llnma3r 39500 cdlema1N 39503 cdlemblem 39505 paddasslem5 39536 paddasslem12 39543 paddasslem13 39544 llnexchb2 39581 dalawlem3 39585 dalawlem5 39587 dalawlem8 39590 dalawlem11 39593 dalawlem12 39594 lhp2lt 39713 lhpexle2lem 39721 lhpexle3lem 39723 4atexlemtlw 39779 4atexlemnclw 39782 lautj 39805 cdlemd3 39912 cdleme3g 39946 cdleme3h 39947 cdleme7d 39958 cdleme11c 39973 cdleme15d 39989 cdleme17b 39999 cdleme19a 40015 cdleme20j 40030 cdleme21c 40039 cdleme22b 40053 cdleme22d 40055 cdleme28a 40082 cdleme35a 40160 cdleme35fnpq 40161 cdleme35b 40162 cdleme35f 40166 cdleme42c 40184 cdleme42i 40195 cdlemf1 40273 cdlemg4c 40324 cdlemg6c 40332 cdlemg8b 40340 cdlemg10 40353 cdlemg11b 40354 cdlemg13a 40363 cdlemg17a 40373 cdlemg18b 40391 cdlemg27a 40404 cdlemg33b0 40413 cdlemg35 40425 cdlemg42 40441 cdlemg46 40447 trljco 40452 tendopltp 40492 cdlemk3 40545 cdlemk10 40555 cdlemk1u 40571 cdlemk39 40628 dialss 40758 dia2dimlem1 40776 dia2dimlem10 40785 dia2dimlem12 40787 cdlemm10N 40830 djajN 40849 diblss 40882 cdlemn2 40907 dihord2pre2 40938 dib2dim 40955 dih2dimb 40956 dih2dimbALTN 40957 dihmeetlem6 41021 dihjatcclem1 41130 |
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