Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatjidm | Structured version Visualization version GIF version |
Description: Idempotence of join operation. Frequently-used special case of latjcom 16882 for atoms. (Contributed by NM, 15-Jul-2012.) |
Ref | Expression |
---|---|
hlatjcom.j | ⊢ ∨ = (join‘𝐾) |
hlatjcom.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlatjidm | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴) → (𝑋 ∨ 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 33668 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | hlatjcom.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 33594 | . 2 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ (Base‘𝐾)) |
5 | hlatjcom.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
6 | 2, 5 | latjidm 16897 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑋) = 𝑋) |
7 | 1, 4, 6 | syl2an 493 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴) → (𝑋 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 joincjn 16767 Latclat 16868 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-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-lat 16869 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 |
This theorem is referenced by: atcvr0eq 33730 lnnat 33731 atcvrj0 33732 atltcvr 33739 3dim2 33772 3dim3 33773 islln2a 33821 2at0mat0 33829 lplnnle2at 33845 lplnnleat 33846 islpln2a 33852 lvolnle3at 33886 lvolnleat 33887 lvolnlelln 33888 2atnelvolN 33891 islvol2aN 33896 dalempnes 33955 dalemqnet 33956 2llnma3r 34092 dalawlem12 34186 4atex2-0aOLDN 34382 idltrn 34454 trl0 34475 trlval3 34492 cdleme3b 34534 cdleme11h 34571 cdleme16c 34585 cdleme18b 34597 cdleme20j 34624 cdleme42ke 34791 cdleme50trn3 34859 cdlemb3 34912 cdlemg8a 34933 trlcone 35034 dia2dimlem13 35383 |
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