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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 2610 | . . 3 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾) | |
3 | polssat.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
4 | 1, 2, 3 | polsubN 34211 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ (PSubSp‘𝐾)) |
5 | 1, 2 | psubssat 34058 | . 2 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ∈ (PSubSp‘𝐾)) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
6 | 4, 5 | syldan 486 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 Atomscatm 33568 HLchlt 33655 PSubSpcpsubsp 33800 ⊥𝑃cpolN 34206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-undef 7286 df-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-psubsp 33807 df-pmap 33808 df-polarityN 34207 |
This theorem is referenced by: 2polcon4bN 34222 polcon2N 34223 pclss2polN 34225 2pmaplubN 34230 paddunN 34231 ispsubcl2N 34251 poml5N 34258 osumcllem1N 34260 osumcllem2N 34261 osumcllem3N 34262 osumcllem9N 34268 osumcllem11N 34270 pexmidN 34273 pexmidlem2N 34275 pexmidlem3N 34276 pexmidlem7N 34280 pexmidlem8N 34281 |
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