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Mirrors > Home > MPE Home > Th. List > Mathboxes > polsubclN | Structured version Visualization version GIF version |
Description: A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polsubcl.a | ⊢ 𝐴 = (Atoms‘𝐾) |
polsubcl.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
polsubcl.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
polsubclN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
2 | eqid 2824 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polsubcl.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2824 | . . 3 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polsubcl.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | polval2N 37046 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋)))) |
7 | hlop 36502 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
8 | 7 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP) |
9 | hlclat 36498 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
10 | eqid 2824 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 3 | atssbase 36430 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
12 | sstr 3978 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
13 | 11, 12 | mpan2 689 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
14 | 10, 1 | clatlubcl 17725 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
15 | 9, 13, 14 | syl2an 597 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
16 | 10, 2 | opoccl 36334 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
17 | 8, 15, 16 | syl2anc 586 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
18 | polsubcl.c | . . . 4 ⊢ 𝐶 = (PSubCl‘𝐾) | |
19 | 10, 4, 18 | pmapsubclN 37086 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝐶) |
20 | 17, 19 | syldan 593 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝐶) |
21 | 6, 20 | eqeltrd 2916 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ‘cfv 6358 Basecbs 16486 occoc 16576 lubclub 17555 CLatccla 17720 OPcops 36312 Atomscatm 36403 HLchlt 36490 pmapcpmap 36637 ⊥𝑃cpolN 37042 PSubClcpscN 37074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-undef 7942 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-pmap 36644 df-polarityN 37043 df-psubclN 37075 |
This theorem is referenced by: osumcllem9N 37104 pexmidN 37109 |
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