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 2737 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
2 | eqid 2737 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polsubsp.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2737 | . . 3 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polsubsp.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | polval2N 37657 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋)))) |
7 | hllat 37114 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ Lat) |
9 | hlop 37113 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP) |
11 | hlclat 37109 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
12 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 3 | atssbase 37041 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
14 | sstr 3909 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
15 | 13, 14 | mpan2 691 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
16 | 12, 1 | clatlubcl 18009 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
17 | 11, 15, 16 | syl2an 599 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
18 | 12, 2 | opoccl 36945 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
19 | 10, 17, 18 | syl2anc 587 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
20 | polsubsp.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
21 | 12, 20, 4 | pmapsub 37519 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝑆) |
22 | 8, 19, 21 | syl2anc 587 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝑆) |
23 | 6, 22 | eqeltrd 2838 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ‘cfv 6380 Basecbs 16760 occoc 16810 lubclub 17816 Latclat 17937 CLatccla 18004 OPcops 36923 Atomscatm 37014 HLchlt 37101 PSubSpcpsubsp 37247 pmapcpmap 37248 ⊥𝑃cpolN 37653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-riotaBAD 36704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-undef 8015 df-proset 17802 df-poset 17820 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p1 17932 df-lat 17938 df-clat 18005 df-oposet 36927 df-ol 36929 df-oml 36930 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-psubsp 37254 df-pmap 37255 df-polarityN 37654 |
This theorem is referenced by: polssatN 37659 pclss2polN 37672 psubclsubN 37691 osumcllem1N 37707 |
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