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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 16884 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 33679 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑃)) |
5 | 4 | 3com23 1263 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑃)) |
6 | 2, 3 | hlatjcom 33672 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
7 | 5, 6 | breqtrrd 4611 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 lecple 15775 joincjn 16767 Atomscatm 33568 HLchlt 33655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-lub 16797 df-join 16799 df-lat 16869 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 |
This theorem is referenced by: 2llnne2N 33712 cvrat3 33746 cvrat4 33747 hlatexch3N 33784 hlatexch4 33785 dalem3 33968 dalem25 34002 lnatexN 34083 lncmp 34087 2llnma3r 34092 paddasslem5 34128 dalawlem3 34177 dalawlem6 34180 dalawlem7 34181 dalawlem12 34186 lhp2atne 34338 lhp2at0ne 34340 4atexlemunv 34370 cdlemc2 34497 cdlemc5 34500 cdleme3h 34540 cdleme7 34554 cdleme9 34558 cdleme11c 34566 cdleme11dN 34567 cdleme11j 34572 cdleme16b 34584 cdleme17b 34592 cdleme18a 34596 cdleme18b 34597 cdleme18c 34598 cdleme20yOLD 34608 cdleme19a 34609 cdleme20d 34618 cdleme20j 34624 cdleme21ct 34635 cdleme22a 34646 cdleme22e 34650 cdleme22eALTN 34651 cdleme35b 34756 cdlemg9a 34938 cdlemg12a 34949 cdlemg13a 34957 cdlemg17a 34967 cdlemg17g 34973 cdlemg18c 34986 cdlemg33b0 35007 cdlemg46 35041 cdlemh1 35121 cdlemh 35123 cdlemk4 35140 cdlemki 35147 cdlemksv2 35153 cdlemk12 35156 cdlemk15 35161 cdlemk12u 35178 cdlemkid1 35228 dia2dimlem1 35371 dia2dimlem3 35373 cdlemn10 35513 dihjatcclem1 35725 |
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