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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 31526 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 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 | lejoin1 18429 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ (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: latjlej1 18498 latnlej 18501 latnlej2 18504 latjidm 18507 latnle 18518 latabs2 18521 latmlej11 18523 latjass 18528 mod1ile 18538 lubun 18560 oldmm1 39218 olj01 39226 omllaw5N 39248 cvlexchb1 39331 cvlsupr2 39344 cvlsupr7 39349 hlatlej1 39376 hlrelat5N 39403 2atjm 39447 2llnmj 39562 lplnexllnN 39566 2llnjaN 39568 2llnm2N 39570 4atlem3a 39599 2lplnja 39621 2lplnm2N 39623 2lplnmj 39624 dalemply 39656 dalemsly 39657 dalem10 39675 dalem13 39678 dalem21 39696 dalem55 39729 2llnma1b 39788 cdlema1N 39793 elpaddn0 39802 paddasslem12 39833 paddasslem13 39834 pmapjoin 39854 dalawlem2 39874 dalawlem7 39879 dalawlem11 39883 dalawlem12 39884 lhpmcvr3 40027 lhpmcvr5N 40029 lhpmcvr6N 40030 lautj 40095 trljat1 40168 cdlemc1 40193 cdlemc4 40196 cdleme1 40229 cdleme8 40252 cdleme11g 40267 cdleme22e 40346 cdleme22eALTN 40347 cdleme23b 40352 cdleme23c 40353 cdleme27N 40371 cdleme30a 40380 cdleme35fnpq 40451 cdleme35b 40452 cdleme35c 40453 cdleme42h 40484 cdleme42i 40485 cdleme48bw 40504 cdlemg2fv2 40602 cdlemg7fvbwN 40609 cdlemg8b 40630 cdlemg11b 40644 trlcolem 40728 trljco 40742 cdlemi1 40820 cdlemk48 40952 cdlemn2 41197 dihjustlem 41218 dihord1 41220 dihord5apre 41264 dihglbcpreN 41302 dihmeetlem3N 41307 dihmeetlem11N 41319 |
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