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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 17658 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 36379 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2818 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | hlatlej.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 36305 | . 2 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
5 | 2, 3 | atbase 36305 | . 2 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
6 | hlatlej.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
7 | hlatlej.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
8 | 2, 6, 7 | latlej1 17658 | . 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 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 joincjn 17542 Latclat 17643 Atomscatm 36279 HLchlt 36366 |
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 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-lub 17572 df-join 17574 df-lat 17644 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 |
This theorem is referenced by: hlatlej2 36392 cvratlem 36437 cvrat4 36459 ps-2 36494 lplnllnneN 36572 dalem1 36675 lnatexN 36795 lncmp 36799 2atm2atN 36801 2llnma3r 36804 dalawlem3 36889 dalawlem6 36892 dalawlem7 36893 dalawlem12 36898 trlval4 37204 cdlemc5 37211 cdlemc6 37212 cdlemd3 37216 cdleme0cp 37230 cdleme3h 37251 cdleme5 37256 cdleme9 37269 cdleme11c 37277 cdleme15b 37291 cdleme17b 37303 cdleme19a 37319 cdleme20c 37327 cdleme20j 37334 cdleme21c 37343 cdleme22b 37357 cdleme22d 37359 cdleme22e 37360 cdleme22eALTN 37361 cdleme35e 37469 cdleme35f 37470 cdleme42a 37487 cdleme17d2 37511 cdlemeg46req 37545 cdlemg13a 37667 cdlemg17a 37677 cdlemg18b 37695 cdlemg27a 37708 trlcoabs2N 37738 cdlemg42 37745 cdlemk4 37850 cdlemk1u 37875 cdlemk39 37932 dia2dimlem1 38080 dia2dimlem2 38081 dia2dimlem3 38082 cdlemm10N 38134 cdlemn10 38222 dihjatcclem1 38434 |
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