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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 2760 | . . 3 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾) | |
3 | polssat.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
4 | 1, 2, 3 | polsubN 35696 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ (PSubSp‘𝐾)) |
5 | 1, 2 | psubssat 35543 | . 2 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ∈ (PSubSp‘𝐾)) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
6 | 4, 5 | syldan 488 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 ‘cfv 6049 Atomscatm 35053 HLchlt 35140 PSubSpcpsubsp 35285 ⊥𝑃cpolN 35691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-riotaBAD 34742 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-undef 7568 df-preset 17129 df-poset 17147 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p1 17241 df-lat 17247 df-clat 17309 df-oposet 34966 df-ol 34968 df-oml 34969 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-psubsp 35292 df-pmap 35293 df-polarityN 35692 |
This theorem is referenced by: 2polcon4bN 35707 polcon2N 35708 pclss2polN 35710 2pmaplubN 35715 paddunN 35716 ispsubcl2N 35736 poml5N 35743 osumcllem1N 35745 osumcllem2N 35746 osumcllem3N 35747 osumcllem9N 35753 osumcllem11N 35755 pexmidN 35758 pexmidlem2N 35760 pexmidlem3N 35761 pexmidlem7N 35765 pexmidlem8N 35766 |
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