Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 17660 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 36381 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | hlatlej.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 36307 | . 2 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
5 | 2, 3 | atbase 36307 | . 2 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
6 | hlatlej.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
7 | hlatlej.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
8 | 2, 6, 7 | latlej1 17660 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
9 | 1, 4, 5, 8 | syl3an 1152 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 lecple 16562 joincjn 17544 Latclat 17645 Atomscatm 36281 HLchlt 36368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-lub 17574 df-join 17576 df-lat 17646 df-ats 36285 df-atl 36316 df-cvlat 36340 df-hlat 36369 |
This theorem is referenced by: hlatlej2 36394 cvratlem 36439 cvrat4 36461 ps-2 36496 lplnllnneN 36574 dalem1 36677 lnatexN 36797 lncmp 36801 2atm2atN 36803 2llnma3r 36806 dalawlem3 36891 dalawlem6 36894 dalawlem7 36895 dalawlem12 36900 trlval4 37206 cdlemc5 37213 cdlemc6 37214 cdlemd3 37218 cdleme0cp 37232 cdleme3h 37253 cdleme5 37258 cdleme9 37271 cdleme11c 37279 cdleme15b 37293 cdleme17b 37305 cdleme19a 37321 cdleme20c 37329 cdleme20j 37336 cdleme21c 37345 cdleme22b 37359 cdleme22d 37361 cdleme22e 37362 cdleme22eALTN 37363 cdleme35e 37471 cdleme35f 37472 cdleme42a 37489 cdleme17d2 37513 cdlemeg46req 37547 cdlemg13a 37669 cdlemg17a 37679 cdlemg18b 37697 cdlemg27a 37710 trlcoabs2N 37740 cdlemg42 37747 cdlemk4 37852 cdlemk1u 37877 cdlemk39 37934 dia2dimlem1 38082 dia2dimlem2 38083 dia2dimlem3 38084 cdlemm10N 38136 cdlemn10 38224 dihjatcclem1 38436 |
Copyright terms: Public domain | W3C validator |