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Mathbox for Norm Megill |
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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 4974 | . 2 ⊢ 𝑋 ⊆ ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦} | |
2 | pclss.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | eqid 2726 | . . 3 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾) | |
4 | pclss.c | . . 3 ⊢ 𝑈 = (PCl‘𝐾) | |
5 | 2, 3, 4 | pclvalN 39589 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦}) |
6 | 1, 5 | sseqtrrid 4033 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ (𝑈‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {crab 3419 ⊆ wss 3947 ∩ cint 4954 ‘cfv 6554 Atomscatm 38961 PSubSpcpsubsp 39195 PClcpclN 39586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-psubsp 39202 df-pclN 39587 |
This theorem is referenced by: pclunN 39597 pcl0bN 39622 pclfinclN 39649 |
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