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Mathbox for Norm Megill |
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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 17662 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 36659 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2798 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | hlatlej.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 36585 | . 2 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
5 | 2, 3 | atbase 36585 | . 2 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
6 | hlatlej.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
7 | hlatlej.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
8 | 2, 6, 7 | latlej1 17662 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
9 | 1, 4, 5, 8 | syl3an 1157 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 joincjn 17546 Latclat 17647 Atomscatm 36559 HLchlt 36646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-lub 17576 df-join 17578 df-lat 17648 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 |
This theorem is referenced by: hlatlej2 36672 cvratlem 36717 cvrat4 36739 ps-2 36774 lplnllnneN 36852 dalem1 36955 lnatexN 37075 lncmp 37079 2atm2atN 37081 2llnma3r 37084 dalawlem3 37169 dalawlem6 37172 dalawlem7 37173 dalawlem12 37178 trlval4 37484 cdlemc5 37491 cdlemc6 37492 cdlemd3 37496 cdleme0cp 37510 cdleme3h 37531 cdleme5 37536 cdleme9 37549 cdleme11c 37557 cdleme15b 37571 cdleme17b 37583 cdleme19a 37599 cdleme20c 37607 cdleme20j 37614 cdleme21c 37623 cdleme22b 37637 cdleme22d 37639 cdleme22e 37640 cdleme22eALTN 37641 cdleme35e 37749 cdleme35f 37750 cdleme42a 37767 cdleme17d2 37791 cdlemeg46req 37825 cdlemg13a 37947 cdlemg17a 37957 cdlemg18b 37975 cdlemg27a 37988 trlcoabs2N 38018 cdlemg42 38025 cdlemk4 38130 cdlemk1u 38155 cdlemk39 38212 dia2dimlem1 38360 dia2dimlem2 38361 dia2dimlem3 38362 cdlemm10N 38414 cdlemn10 38502 dihjatcclem1 38714 |
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