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 485 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ HL) | |
2 | pclss2pol.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | pclss2pol.o | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
4 | 2, 3 | 2polssN 37084 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
5 | 2, 3 | polssatN 37077 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
6 | 2, 3 | polssatN 37077 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝐴) |
7 | 5, 6 | syldan 593 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝐴) |
8 | pclss2pol.c | . . . 4 ⊢ 𝑈 = (PCl‘𝐾) | |
9 | 2, 8 | pclssN 37063 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘( ⊥ ‘( ⊥ ‘𝑋)))) |
10 | 1, 4, 7, 9 | syl3anc 1366 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘( ⊥ ‘( ⊥ ‘𝑋)))) |
11 | eqid 2820 | . . . . 5 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾) | |
12 | 2, 11, 3 | polsubN 37076 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ (PSubSp‘𝐾)) |
13 | 5, 12 | syldan 593 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ (PSubSp‘𝐾)) |
14 | 11, 8 | pclidN 37065 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ (PSubSp‘𝐾)) → (𝑈‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘( ⊥ ‘𝑋))) |
15 | 13, 14 | syldan 593 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘( ⊥ ‘𝑋))) |
16 | 10, 15 | sseqtrd 4000 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 ‘cfv 6348 Atomscatm 36432 HLchlt 36519 PSubSpcpsubsp 36665 PClcpclN 37056 ⊥𝑃cpolN 37071 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-riotaBAD 36122 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-undef 7932 df-proset 17533 df-poset 17551 df-plt 17563 df-lub 17579 df-glb 17580 df-join 17581 df-meet 17582 df-p0 17644 df-p1 17645 df-lat 17651 df-clat 17713 df-oposet 36345 df-ol 36347 df-oml 36348 df-covers 36435 df-ats 36436 df-atl 36467 df-cvlat 36491 df-hlat 36520 df-psubsp 36672 df-pmap 36673 df-pclN 37057 df-polarityN 37072 |
This theorem is referenced by: pcl0N 37091 |
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