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 17663 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 36493 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | hlatjcom.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 36419 | . 2 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ (Base‘𝐾)) |
5 | hlatjcom.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
6 | 2, 5 | latjidm 17678 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑋) = 𝑋) |
7 | 1, 4, 6 | syl2an 597 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴) → (𝑋 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 joincjn 17548 Latclat 17649 Atomscatm 36393 HLchlt 36480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-lat 17650 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 |
This theorem is referenced by: atcvr0eq 36556 lnnat 36557 atcvrj0 36558 atltcvr 36565 3dim2 36598 3dim3 36599 islln2a 36647 2at0mat0 36655 lplnnle2at 36671 lplnnleat 36672 islpln2a 36678 lvolnle3at 36712 lvolnleat 36713 lvolnlelln 36714 2atnelvolN 36717 islvol2aN 36722 dalempnes 36781 dalemqnet 36782 2llnma3r 36918 dalawlem12 37012 4atex2-0aOLDN 37208 idltrn 37280 trl0 37300 trlval3 37317 cdleme3b 37359 cdleme11h 37396 cdleme16c 37410 cdleme18b 37422 cdleme20j 37448 cdleme42ke 37615 cdleme50trn3 37683 cdlemb3 37736 cdlemg8a 37757 trlcone 37858 dia2dimlem13 38206 |
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