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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 2726 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
2 | eqid 2726 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polsubsp.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2726 | . . 3 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polsubsp.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | polval2N 39605 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋)))) |
7 | hllat 39061 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
8 | 7 | adantr 479 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ Lat) |
9 | hlop 39060 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | 9 | adantr 479 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP) |
11 | hlclat 39056 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
12 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 3 | atssbase 38988 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
14 | sstr 3988 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
15 | 13, 14 | mpan2 689 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
16 | 12, 1 | clatlubcl 18528 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
17 | 11, 15, 16 | syl2an 594 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
18 | 12, 2 | opoccl 38892 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
19 | 10, 17, 18 | syl2anc 582 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
20 | polsubsp.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
21 | 12, 20, 4 | pmapsub 39467 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝑆) |
22 | 8, 19, 21 | syl2anc 582 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝑆) |
23 | 6, 22 | eqeltrd 2826 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6554 Basecbs 17213 occoc 17274 lubclub 18334 Latclat 18456 CLatccla 18523 OPcops 38870 Atomscatm 38961 HLchlt 39048 PSubSpcpsubsp 39195 pmapcpmap 39196 ⊥𝑃cpolN 39601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-proset 18320 df-poset 18338 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p1 18451 df-lat 18457 df-clat 18524 df-oposet 38874 df-ol 38876 df-oml 38877 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-psubsp 39202 df-pmap 39203 df-polarityN 39602 |
This theorem is referenced by: polssatN 39607 pclss2polN 39620 psubclsubN 39639 osumcllem1N 39655 |
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