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Mirrors > Home > MPE Home > Th. List > Mathboxes > pclidN | Structured version Visualization version GIF version |
Description: The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclid.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
pclid.c | ⊢ 𝑈 = (PCl‘𝐾) |
Ref | Expression |
---|---|
pclidN | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | pclid.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
3 | 1, 2 | psubssat 36770 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾)) |
4 | pclid.c | . . . 4 ⊢ 𝑈 = (PCl‘𝐾) | |
5 | 1, 2, 4 | pclvalN 36906 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
6 | 3, 5 | syldan 591 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
7 | intmin 4887 | . . 3 ⊢ (𝑋 ∈ 𝑆 → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} = 𝑋) | |
8 | 7 | adantl 482 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} = 𝑋) |
9 | 6, 8 | eqtrd 2853 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 ∩ cint 4867 ‘cfv 6348 Atomscatm 36279 PSubSpcpsubsp 36512 PClcpclN 36903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 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-ov 7148 df-psubsp 36519 df-pclN 36904 |
This theorem is referenced by: pclbtwnN 36913 pclunN 36914 pclun2N 36915 pclfinN 36916 pclss2polN 36937 pclfinclN 36966 |
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