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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatlej2 | Structured version Visualization version GIF version | ||
| Description: A join's second argument is less than or equal to the join. Special case of latlej2 18494 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 |
|---|---|
| hlatlej2 | ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlatlej.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 2 | hlatlej.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
| 3 | hlatlej.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 4 | 1, 2, 3 | hlatlej1 39376 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑃)) |
| 5 | 4 | 3com23 1127 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑃)) |
| 6 | 2, 3 | hlatjcom 39369 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
| 7 | 5, 6 | breqtrrd 5171 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (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 lecple 17304 joincjn 18357 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: 2llnne2N 39410 cvrat3 39444 cvrat4 39445 hlatexch3N 39482 hlatexch4 39483 dalem3 39666 dalem25 39700 lnatexN 39781 lncmp 39785 2llnma3r 39790 paddasslem5 39826 dalawlem3 39875 dalawlem6 39878 dalawlem7 39879 dalawlem12 39884 lhp2atne 40036 lhp2at0ne 40038 4atexlemunv 40068 cdlemc2 40194 cdlemc5 40197 cdleme3h 40237 cdleme7 40251 cdleme9 40255 cdleme11c 40263 cdleme11dN 40264 cdleme11j 40269 cdleme16b 40281 cdleme17b 40289 cdleme18a 40293 cdleme18b 40294 cdleme18c 40295 cdleme19a 40305 cdleme20d 40314 cdleme20j 40320 cdleme21ct 40331 cdleme22a 40342 cdleme22e 40346 cdleme22eALTN 40347 cdleme35b 40452 cdlemg9a 40634 cdlemg12a 40645 cdlemg13a 40653 cdlemg17a 40663 cdlemg17g 40669 cdlemg18c 40682 cdlemg33b0 40703 cdlemg46 40737 cdlemh1 40817 cdlemh 40819 cdlemk4 40836 cdlemki 40843 cdlemksv2 40849 cdlemk12 40852 cdlemk15 40857 cdlemk12u 40874 cdlemkid1 40924 dia2dimlem1 41066 dia2dimlem3 41068 cdlemn10 41208 dihjatcclem1 41420 |
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