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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatlej1 | Structured version Visualization version GIF version | ||
| Description: A join's first argument is less than or equal to the join. Special case of latlej1 18493 to show an atom is on a line. (Contributed by NM, 15-May-2013.) |
| Ref | Expression |
|---|---|
| hlatlej.l | ⊢ ≤ = (le‘𝐾) |
| hlatlej.j | ⊢ ∨ = (join‘𝐾) |
| hlatlej.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| Ref | Expression |
|---|---|
| hlatlej1 | ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hllat 39364 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
| 2 | eqid 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 3 | hlatlej.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 4 | 2, 3 | atbase 39290 | . 2 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
| 5 | 2, 3 | atbase 39290 | . 2 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
| 6 | hlatlej.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 7 | hlatlej.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
| 8 | 2, 6, 7 | latlej1 18493 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
| 9 | 1, 4, 5, 8 | syl3an 1161 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 Latclat 18476 Atomscatm 39264 HLchlt 39351 |
| 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 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 |
| This theorem is referenced by: hlatlej2 39377 cvratlem 39423 cvrat4 39445 ps-2 39480 lplnllnneN 39558 dalem1 39661 lnatexN 39781 lncmp 39785 2atm2atN 39787 2llnma3r 39790 dalawlem3 39875 dalawlem6 39878 dalawlem7 39879 dalawlem12 39884 trlval4 40190 cdlemc5 40197 cdlemc6 40198 cdlemd3 40202 cdleme0cp 40216 cdleme3h 40237 cdleme5 40242 cdleme9 40255 cdleme11c 40263 cdleme15b 40277 cdleme17b 40289 cdleme19a 40305 cdleme20c 40313 cdleme20j 40320 cdleme21c 40329 cdleme22b 40343 cdleme22d 40345 cdleme22e 40346 cdleme22eALTN 40347 cdleme35e 40455 cdleme35f 40456 cdleme42a 40473 cdleme17d2 40497 cdlemeg46req 40531 cdlemg13a 40653 cdlemg17a 40663 cdlemg18b 40681 cdlemg27a 40694 trlcoabs2N 40724 cdlemg42 40731 cdlemk4 40836 cdlemk1u 40861 cdlemk39 40918 dia2dimlem1 41066 dia2dimlem2 41067 dia2dimlem3 41068 cdlemm10N 41120 cdlemn10 41208 dihjatcclem1 41420 |
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