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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 2polssN | Structured version Visualization version GIF version | ||
| Description: A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| 2polss.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| 2polss.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
| Ref | Expression |
|---|---|
| 2polssN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlclat 39359 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
| 2 | 1 | ad3antrrr 730 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝐾 ∈ CLat) |
| 3 | simpr 484 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝑋) | |
| 4 | simpllr 776 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑋 ⊆ 𝐴) | |
| 5 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 6 | 2polss.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 7 | 5, 6 | atssbase 39291 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
| 8 | 4, 7 | sstrdi 3996 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐾)) |
| 9 | eqid 2737 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 10 | eqid 2737 | . . . . . 6 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
| 11 | 5, 9, 10 | lubel 18559 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑝 ∈ 𝑋 ∧ 𝑋 ⊆ (Base‘𝐾)) → 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)) |
| 12 | 2, 3, 8, 11 | syl3anc 1373 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)) |
| 13 | 12 | ex 412 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴) → (𝑝 ∈ 𝑋 → 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋))) |
| 14 | 13 | ss2rabdv 4076 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)}) |
| 15 | sseqin2 4223 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑋) = 𝑋) | |
| 16 | 15 | biimpi 216 | . . . 4 ⊢ (𝑋 ⊆ 𝐴 → (𝐴 ∩ 𝑋) = 𝑋) |
| 17 | 16 | adantl 481 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝐴 ∩ 𝑋) = 𝑋) |
| 18 | dfin5 3959 | . . 3 ⊢ (𝐴 ∩ 𝑋) = {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋} | |
| 19 | 17, 18 | eqtr3di 2792 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋}) |
| 20 | eqid 2737 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
| 21 | 2polss.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
| 22 | 10, 6, 20, 21 | 2polvalN 39916 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋))) |
| 23 | sstr 3992 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
| 24 | 7, 23 | mpan2 691 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
| 25 | 5, 10 | clatlubcl 18548 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
| 26 | 1, 24, 25 | syl2an 596 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
| 27 | 5, 9, 6, 20 | pmapval 39759 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋)) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)}) |
| 28 | 26, 27 | syldan 591 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋)) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)}) |
| 29 | 22, 28 | eqtrd 2777 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)}) |
| 30 | 14, 19, 29 | 3sstr4d 4039 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 lubclub 18355 CLatccla 18543 Atomscatm 39264 HLchlt 39351 pmapcpmap 39499 ⊥𝑃cpolN 39904 |
| 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-iin 4994 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-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-pmap 39506 df-polarityN 39905 |
| This theorem is referenced by: polcon2N 39921 pclss2polN 39923 sspmaplubN 39927 paddunN 39929 pnonsingN 39935 osumcllem1N 39958 osumcllem11N 39968 pexmidN 39971 |
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