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Mirrors > Home > MPE Home > Th. List > Mathboxes > pclssidN | Structured version Visualization version GIF version |
Description: A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclss.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pclss.c | ⊢ 𝑈 = (PCl‘𝐾) |
Ref | Expression |
---|---|
pclssidN | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ (𝑈‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintub 4893 | . 2 ⊢ 𝑋 ⊆ ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦} | |
2 | pclss.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | eqid 2821 | . . 3 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾) | |
4 | pclss.c | . . 3 ⊢ 𝑈 = (PCl‘𝐾) | |
5 | 2, 3, 4 | pclvalN 37025 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦}) |
6 | 1, 5 | sseqtrrid 4019 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ (𝑈‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 ∩ cint 4875 ‘cfv 6354 Atomscatm 36398 PSubSpcpsubsp 36631 PClcpclN 37022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-psubsp 36638 df-pclN 37023 |
This theorem is referenced by: pclunN 37033 pcl0bN 37058 pclfinclN 37085 |
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