| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pclss2polN | Structured version Visualization version GIF version | ||
| Description: The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| pclss2pol.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| pclss2pol.o | ⊢ ⊥ = (⊥𝑃‘𝐾) |
| pclss2pol.c | ⊢ 𝑈 = (PCl‘𝐾) |
| Ref | Expression |
|---|---|
| pclss2polN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 482 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ HL) | |
| 2 | pclss2pol.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 3 | pclss2pol.o | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
| 4 | 2, 3 | 2polssN 39917 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
| 5 | 2, 3 | polssatN 39910 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
| 6 | 2, 3 | polssatN 39910 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝐴) |
| 7 | 5, 6 | syldan 591 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝐴) |
| 8 | pclss2pol.c | . . . 4 ⊢ 𝑈 = (PCl‘𝐾) | |
| 9 | 2, 8 | pclssN 39896 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘( ⊥ ‘( ⊥ ‘𝑋)))) |
| 10 | 1, 4, 7, 9 | syl3anc 1373 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘( ⊥ ‘( ⊥ ‘𝑋)))) |
| 11 | eqid 2737 | . . . . 5 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾) | |
| 12 | 2, 11, 3 | polsubN 39909 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ (PSubSp‘𝐾)) |
| 13 | 5, 12 | syldan 591 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ (PSubSp‘𝐾)) |
| 14 | 11, 8 | pclidN 39898 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ (PSubSp‘𝐾)) → (𝑈‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘( ⊥ ‘𝑋))) |
| 15 | 13, 14 | syldan 591 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘( ⊥ ‘𝑋))) |
| 16 | 10, 15 | sseqtrd 4020 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 Atomscatm 39264 HLchlt 39351 PSubSpcpsubsp 39498 PClcpclN 39889 ⊥𝑃cpolN 39904 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-psubsp 39505 df-pmap 39506 df-pclN 39890 df-polarityN 39905 |
| This theorem is referenced by: pcl0N 39924 |
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