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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > polssatN | Structured version Visualization version GIF version |
Description: The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polssat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
polssat.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
polssatN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | polssat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | eqid 2798 | . . 3 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾) | |
3 | polssat.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
4 | 1, 2, 3 | polsubN 37203 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ (PSubSp‘𝐾)) |
5 | 1, 2 | psubssat 37050 | . 2 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ∈ (PSubSp‘𝐾)) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
6 | 4, 5 | syldan 594 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ‘cfv 6324 Atomscatm 36559 HLchlt 36646 PSubSpcpsubsp 36792 ⊥𝑃cpolN 37198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-undef 7922 df-proset 17530 df-poset 17548 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p1 17642 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-psubsp 36799 df-pmap 36800 df-polarityN 37199 |
This theorem is referenced by: 2polcon4bN 37214 polcon2N 37215 pclss2polN 37217 2pmaplubN 37222 paddunN 37223 ispsubcl2N 37243 poml5N 37250 osumcllem1N 37252 osumcllem2N 37253 osumcllem3N 37254 osumcllem9N 37260 osumcllem11N 37262 pexmidN 37265 pexmidlem2N 37267 pexmidlem3N 37268 pexmidlem7N 37272 pexmidlem8N 37273 |
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