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 17668 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 36498 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | hlatjcom.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 36424 | . 2 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ (Base‘𝐾)) |
5 | hlatjcom.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
6 | 2, 5 | latjidm 17683 | . 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 6354 (class class class)co 7155 Basecbs 16482 joincjn 17553 Latclat 17654 Atomscatm 36398 HLchlt 36485 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-proset 17537 df-poset 17555 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-lat 17655 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 |
This theorem is referenced by: atcvr0eq 36561 lnnat 36562 atcvrj0 36563 atltcvr 36570 3dim2 36603 3dim3 36604 islln2a 36652 2at0mat0 36660 lplnnle2at 36676 lplnnleat 36677 islpln2a 36683 lvolnle3at 36717 lvolnleat 36718 lvolnlelln 36719 2atnelvolN 36722 islvol2aN 36727 dalempnes 36786 dalemqnet 36787 2llnma3r 36923 dalawlem12 37017 4atex2-0aOLDN 37213 idltrn 37285 trl0 37305 trlval3 37322 cdleme3b 37364 cdleme11h 37401 cdleme16c 37415 cdleme18b 37427 cdleme20j 37453 cdleme42ke 37620 cdleme50trn3 37688 cdlemb3 37741 cdlemg8a 37762 trlcone 37863 dia2dimlem13 38211 |
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