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Mathbox for Norm Megill |
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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 17663 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 36671 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑃)) |
5 | 4 | 3com23 1123 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑃)) |
6 | 2, 3 | hlatjcom 36664 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
7 | 5, 6 | breqtrrd 5058 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (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 lecple 16564 joincjn 17546 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: 2llnne2N 36704 cvrat3 36738 cvrat4 36739 hlatexch3N 36776 hlatexch4 36777 dalem3 36960 dalem25 36994 lnatexN 37075 lncmp 37079 2llnma3r 37084 paddasslem5 37120 dalawlem3 37169 dalawlem6 37172 dalawlem7 37173 dalawlem12 37178 lhp2atne 37330 lhp2at0ne 37332 4atexlemunv 37362 cdlemc2 37488 cdlemc5 37491 cdleme3h 37531 cdleme7 37545 cdleme9 37549 cdleme11c 37557 cdleme11dN 37558 cdleme11j 37563 cdleme16b 37575 cdleme17b 37583 cdleme18a 37587 cdleme18b 37588 cdleme18c 37589 cdleme19a 37599 cdleme20d 37608 cdleme20j 37614 cdleme21ct 37625 cdleme22a 37636 cdleme22e 37640 cdleme22eALTN 37641 cdleme35b 37746 cdlemg9a 37928 cdlemg12a 37939 cdlemg13a 37947 cdlemg17a 37957 cdlemg17g 37963 cdlemg18c 37976 cdlemg33b0 37997 cdlemg46 38031 cdlemh1 38111 cdlemh 38113 cdlemk4 38130 cdlemki 38137 cdlemksv2 38143 cdlemk12 38146 cdlemk15 38151 cdlemk12u 38168 cdlemkid1 38218 dia2dimlem1 38360 dia2dimlem3 38362 cdlemn10 38502 dihjatcclem1 38714 |
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