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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 17060 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 34650 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2622 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | hlatlej.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 34576 | . 2 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
5 | 2, 3 | atbase 34576 | . 2 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
6 | hlatlej.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
7 | hlatlej.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
8 | 2, 6, 7 | latlej1 17060 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
9 | 1, 4, 5, 8 | syl3an 1368 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 joincjn 16944 Latclat 17045 Atomscatm 34550 HLchlt 34637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-lub 16974 df-join 16976 df-lat 17046 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 |
This theorem is referenced by: hlatlej2 34662 cvratlem 34707 cvrat4 34729 ps-2 34764 lplnllnneN 34842 dalem1 34945 lnatexN 35065 lncmp 35069 2atm2atN 35071 2llnma3r 35074 dalawlem3 35159 dalawlem6 35162 dalawlem7 35163 dalawlem12 35168 trlval4 35475 cdlemc5 35482 cdlemc6 35483 cdlemd3 35487 cdleme0cp 35501 cdleme3h 35522 cdleme5 35527 cdleme9 35540 cdleme11c 35548 cdleme15b 35562 cdleme17b 35574 cdleme19a 35591 cdleme20c 35599 cdleme20j 35606 cdleme21c 35615 cdleme22b 35629 cdleme22d 35631 cdleme22e 35632 cdleme22eALTN 35633 cdleme35e 35741 cdleme35f 35742 cdleme42a 35759 cdleme17d2 35783 cdlemeg46req 35817 cdlemg13a 35939 cdlemg17a 35949 cdlemg18b 35967 cdlemg27a 35980 trlcoabs2N 36010 cdlemg42 36017 cdlemk4 36122 cdlemk1u 36147 cdlemk39 36204 dia2dimlem1 36353 dia2dimlem2 36354 dia2dimlem3 36355 cdlemm10N 36407 cdlemn10 36495 dihjatcclem1 36707 |
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