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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > polsubN | Structured version Visualization version GIF version |
Description: The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polsubsp.a | ⊢ 𝐴 = (Atoms‘𝐾) |
polsubsp.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
polsubsp.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
polsubN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
2 | eqid 2735 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polsubsp.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2735 | . . 3 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polsubsp.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | polval2N 39889 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋)))) |
7 | hllat 39345 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ Lat) |
9 | hlop 39344 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP) |
11 | hlclat 39340 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
12 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 3 | atssbase 39272 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
14 | sstr 4004 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
15 | 13, 14 | mpan2 691 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
16 | 12, 1 | clatlubcl 18561 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
17 | 11, 15, 16 | syl2an 596 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
18 | 12, 2 | opoccl 39176 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
19 | 10, 17, 18 | syl2anc 584 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
20 | polsubsp.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
21 | 12, 20, 4 | pmapsub 39751 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝑆) |
22 | 8, 19, 21 | syl2anc 584 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝑆) |
23 | 6, 22 | eqeltrd 2839 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 Basecbs 17245 occoc 17306 lubclub 18367 Latclat 18489 CLatccla 18556 OPcops 39154 Atomscatm 39245 HLchlt 39332 PSubSpcpsubsp 39479 pmapcpmap 39480 ⊥𝑃cpolN 39885 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18352 df-poset 18371 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p1 18484 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-psubsp 39486 df-pmap 39487 df-polarityN 39886 |
This theorem is referenced by: polssatN 39891 pclss2polN 39904 psubclsubN 39923 osumcllem1N 39939 |
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