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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 2795 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
2 | eqid 2795 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polsubsp.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2795 | . . 3 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polsubsp.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | polval2N 36592 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋)))) |
7 | hllat 36049 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ Lat) |
9 | hlop 36048 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP) |
11 | hlclat 36044 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
12 | eqid 2795 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 3 | atssbase 35976 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
14 | sstr 3897 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
15 | 13, 14 | mpan2 687 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
16 | 12, 1 | clatlubcl 17551 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
17 | 11, 15, 16 | syl2an 595 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
18 | 12, 2 | opoccl 35880 | . . . 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 36454 | . . 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 2883 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ⊆ wss 3859 ‘cfv 6225 Basecbs 16312 occoc 16402 lubclub 17381 Latclat 17484 CLatccla 17546 OPcops 35858 Atomscatm 35949 HLchlt 36036 PSubSpcpsubsp 36182 pmapcpmap 36183 ⊥𝑃cpolN 36588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-riotaBAD 35639 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-undef 7790 df-proset 17367 df-poset 17385 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p1 17479 df-lat 17485 df-clat 17547 df-oposet 35862 df-ol 35864 df-oml 35865 df-ats 35953 df-atl 35984 df-cvlat 36008 df-hlat 36037 df-psubsp 36189 df-pmap 36190 df-polarityN 36589 |
This theorem is referenced by: polssatN 36594 pclss2polN 36607 psubclsubN 36626 osumcllem1N 36642 |
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