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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 2771 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
2 | eqid 2771 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polsubcl.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2771 | . . 3 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polsubcl.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | polval2N 35710 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋)))) |
7 | hlop 35167 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
8 | 7 | adantr 466 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ OP) |
9 | hlclat 35163 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
10 | eqid 2771 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 3 | atssbase 35095 | . . . . . 6 ⊢ 𝐴 ⊆ (Base‘𝐾) |
12 | sstr 3760 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ (Base‘𝐾)) → 𝑋 ⊆ (Base‘𝐾)) | |
13 | 11, 12 | mpan2 663 | . . . . 5 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ⊆ (Base‘𝐾)) |
14 | 10, 1 | clatlubcl 17319 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
15 | 9, 13, 14 | syl2an 575 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) |
16 | 10, 2 | opoccl 34999 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
17 | 8, 15, 16 | syl2anc 565 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) |
18 | polsubcl.c | . . . 4 ⊢ 𝐶 = (PSubCl‘𝐾) | |
19 | 10, 4, 18 | pmapsubclN 35750 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝐶) |
20 | 17, 19 | syldan 571 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∈ 𝐶) |
21 | 6, 20 | eqeltrd 2850 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ‘cfv 6031 Basecbs 16063 occoc 16156 lubclub 17149 CLatccla 17314 OPcops 34977 Atomscatm 35068 HLchlt 35155 pmapcpmap 35301 ⊥𝑃cpolN 35706 PSubClcpscN 35738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-riotaBAD 34757 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-undef 7550 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-p1 17247 df-lat 17253 df-clat 17315 df-oposet 34981 df-ol 34983 df-oml 34984 df-covers 35071 df-ats 35072 df-atl 35103 df-cvlat 35127 df-hlat 35156 df-pmap 35308 df-polarityN 35707 df-psubclN 35739 |
This theorem is referenced by: osumcllem9N 35768 pexmidN 35773 |
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