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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatjcom | Structured version Visualization version GIF version | ||
| Description: Commutatitivity of join operation. Frequently-used special case of latjcom 18492 for atoms. (Contributed by NM, 15-Jun-2012.) |
| Ref | Expression |
|---|---|
| hlatjcom.j | ⊢ ∨ = (join‘𝐾) |
| hlatjcom.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| Ref | Expression |
|---|---|
| hlatjcom | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 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 | 2, 3 | atbase 39290 | . 2 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ (Base‘𝐾)) |
| 6 | hlatjcom.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
| 7 | 2, 6 | latjcom 18492 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋)) |
| 8 | 1, 4, 5, 7 | syl3an 1161 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = 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-lub 18391 df-join 18393 df-lat 18477 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 |
| This theorem is referenced by: hlatj12 39372 hlatjrot 39374 hlatlej2 39377 atbtwnex 39450 3noncolr2 39451 hlatcon2 39454 3dimlem2 39461 3dimlem3 39463 3dimlem3OLDN 39464 3dimlem4 39466 3dimlem4OLDN 39467 ps-1 39479 hlatexch4 39483 lplnribN 39553 4atlem10 39608 4atlem11 39611 dalemswapyz 39658 dalem-cly 39673 dalemswapyzps 39692 dalem24 39699 dalem25 39700 dalem44 39718 2llnma1 39789 2llnma3r 39790 2llnma2rN 39792 llnexchb2 39871 dalawlem4 39876 dalawlem5 39877 dalawlem9 39881 dalawlem11 39883 dalawlem12 39884 dalawlem15 39887 4atexlemex2 40073 4atexlemcnd 40074 ltrncnv 40148 trlcnv 40167 cdlemc6 40198 cdleme7aa 40244 cdleme12 40273 cdleme15a 40276 cdleme15c 40278 cdleme17c 40290 cdlemeda 40300 cdleme19a 40305 cdleme19e 40309 cdleme20bN 40312 cdleme20g 40317 cdleme20m 40325 cdleme21c 40329 cdleme22f 40348 cdleme22g 40350 cdleme35b 40452 cdleme35f 40456 cdleme37m 40464 cdleme39a 40467 cdleme42h 40484 cdleme43aN 40491 cdleme43bN 40492 cdleme43dN 40494 cdleme46f2g2 40495 cdleme46f2g1 40496 cdlemeg46c 40515 cdlemeg46nlpq 40519 cdlemeg46ngfr 40520 cdlemeg46rgv 40530 cdlemeg46gfv 40532 cdlemg2kq 40604 cdlemg4a 40610 cdlemg4d 40615 cdlemg4 40619 cdlemg8c 40631 cdlemg11aq 40640 cdlemg10a 40642 cdlemg12g 40651 cdlemg12 40652 cdlemg13 40654 cdlemg17pq 40674 cdlemg18b 40681 cdlemg18c 40682 cdlemg19 40686 cdlemg21 40688 cdlemk7 40850 cdlemk7u 40872 cdlemkfid1N 40923 dia2dimlem1 41066 dia2dimlem3 41068 dihjatcclem3 41422 dihjat 41425 |
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