| Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
| 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 18492 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 39364 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
| 2 | eqid 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 3 | hlatjcom.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 4 | 2, 3 | atbase 39290 | . 2 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ (Base‘𝐾)) |
| 5 | hlatjcom.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
| 6 | 2, 5 | latjidm 18507 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑋) = 𝑋) |
| 7 | 1, 4, 6 | syl2an 596 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴) → (𝑋 ∨ 𝑋) = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 joincjn 18357 Latclat 18476 Atomscatm 39264 HLchlt 39351 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18340 df-poset 18359 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-lat 18477 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 |
| This theorem is referenced by: atcvr0eq 39428 lnnat 39429 atcvrj0 39430 atltcvr 39437 3dim2 39470 3dim3 39471 islln2a 39519 2at0mat0 39527 lplnnle2at 39543 lplnnleat 39544 islpln2a 39550 lvolnle3at 39584 lvolnleat 39585 lvolnlelln 39586 2atnelvolN 39589 islvol2aN 39594 dalempnes 39653 dalemqnet 39654 2llnma3r 39790 dalawlem12 39884 4atex2-0aOLDN 40080 idltrn 40152 trl0 40172 trlval3 40189 cdleme3b 40231 cdleme11h 40268 cdleme16c 40282 cdleme18b 40294 cdleme20j 40320 cdleme42ke 40487 cdleme50trn3 40555 cdlemb3 40608 cdlemg8a 40629 trlcone 40730 dia2dimlem13 41078 |
| Copyright terms: Public domain | W3C validator |