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| Mirrors > Home > MPE Home > Th. List > latlej2 | Structured version Visualization version GIF version | ||
| Description: A join's second argument is less than or equal to the join. (chub2 31527 analog.) (Contributed by NM, 17-Sep-2011.) |
| Ref | Expression |
|---|---|
| latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
| latlej.l | ⊢ ≤ = (le‘𝐾) |
| latlej.j | ⊢ ∨ = (join‘𝐾) |
| Ref | Expression |
|---|---|
| latlej2 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (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 1137 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
| 5 | simp2 1138 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
| 6 | simp3 1139 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
| 7 | eqid 2737 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
| 8 | 1, 3, 7, 4, 5, 6 | latcl2 18481 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (〈𝑋, 𝑌〉 ∈ dom ∨ ∧ 〈𝑋, 𝑌〉 ∈ dom (meet‘𝐾))) |
| 9 | 8 | simpld 494 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
| 10 | 1, 2, 3, 4, 5, 6, 9 | lejoin2 18430 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ 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 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-lub 18391 df-join 18393 df-lat 18477 |
| This theorem is referenced by: latleeqj1 18496 latjlej1 18498 latnlej 18501 latnlej2 18504 latjass 18528 lubun 18560 oldmm1 39218 cmtcomlemN 39249 cmtbr4N 39256 cvlexchb1 39331 cvlatexch1 39337 cvrval5 39417 2llnjaN 39568 4atlem3b 39600 2lplnja 39621 dalem5 39669 dalem17 39682 dalem39 39713 dalem43 39717 elpaddn0 39802 pmapjoin 39854 dalawlem2 39874 dalawlem11 39883 dalawlem12 39884 lautj 40095 trljat2 40169 cdleme0cq 40217 cdleme1 40229 cdleme3 40239 cdleme5 40242 cdleme7ga 40250 cdleme10 40256 cdleme15b 40277 cdleme16b 40281 cdleme20k 40321 cdleme22e 40346 cdleme22eALTN 40347 cdleme23c 40353 cdleme28a 40372 cdleme32e 40447 cdleme35a 40450 cdlemg4c 40614 cdlemg6c 40622 trlcolem 40728 cdlemi1 40820 dia2dimlem2 41067 cdlemm10N 41120 dihord2pre2 41228 dihord5apre 41264 dihjatc1 41313 |
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