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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 18467 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 39074 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2726 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | hlatjcom.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 39000 | . 2 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ (Base‘𝐾)) |
5 | hlatjcom.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
6 | 2, 5 | latjidm 18482 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑋) = 𝑋) |
7 | 1, 4, 6 | syl2an 594 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴) → (𝑋 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 joincjn 18331 Latclat 18451 Atomscatm 38974 HLchlt 39061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-proset 18315 df-poset 18333 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-lat 18452 df-ats 38978 df-atl 39009 df-cvlat 39033 df-hlat 39062 |
This theorem is referenced by: atcvr0eq 39138 lnnat 39139 atcvrj0 39140 atltcvr 39147 3dim2 39180 3dim3 39181 islln2a 39229 2at0mat0 39237 lplnnle2at 39253 lplnnleat 39254 islpln2a 39260 lvolnle3at 39294 lvolnleat 39295 lvolnlelln 39296 2atnelvolN 39299 islvol2aN 39304 dalempnes 39363 dalemqnet 39364 2llnma3r 39500 dalawlem12 39594 4atex2-0aOLDN 39790 idltrn 39862 trl0 39882 trlval3 39899 cdleme3b 39941 cdleme11h 39978 cdleme16c 39992 cdleme18b 40004 cdleme20j 40030 cdleme42ke 40197 cdleme50trn3 40265 cdlemb3 40318 cdlemg8a 40339 trlcone 40440 dia2dimlem13 40788 |
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