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Mathbox for Norm Megill |
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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 39339 | . . . . . 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 2734 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 2polss.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 5, 6 | atssbase 39271 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
8 | 4, 7 | sstrdi 4007 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐾)) |
9 | eqid 2734 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | eqid 2734 | . . . . . 6 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
11 | 5, 9, 10 | lubel 18571 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑝 ∈ 𝑋 ∧ 𝑋 ⊆ (Base‘𝐾)) → 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)) |
12 | 2, 3, 8, 11 | syl3anc 1370 | . . . 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 4085 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)}) |
15 | sseqin2 4230 | . . . . 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 3970 | . . 3 ⊢ (𝐴 ∩ 𝑋) = {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋} | |
19 | 17, 18 | eqtr3di 2789 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋}) |
20 | eqid 2734 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
21 | 2polss.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
22 | 10, 6, 20, 21 | 2polvalN 39896 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋))) |
23 | sstr 4003 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
24 | 7, 23 | mpan2 691 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
25 | 5, 10 | clatlubcl 18560 | . . . . 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 39739 | . . . 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 2774 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)((lub‘𝐾)‘𝑋)}) |
30 | 14, 19, 29 | 3sstr4d 4042 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 ∩ cin 3961 ⊆ wss 3962 class class class wbr 5147 ‘cfv 6562 Basecbs 17244 lecple 17304 lubclub 18366 CLatccla 18555 Atomscatm 39244 HLchlt 39331 pmapcpmap 39479 ⊥𝑃cpolN 39884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-pmap 39486 df-polarityN 39885 |
This theorem is referenced by: polcon2N 39901 pclss2polN 39903 sspmaplubN 39907 paddunN 39909 pnonsingN 39915 osumcllem1N 39938 osumcllem11N 39948 pexmidN 39951 |
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