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Mirrors > Home > MPE Home > Th. List > Mathboxes > pclfvalN | Structured version Visualization version GIF version |
Description: The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pclfval.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
pclfval.c | ⊢ 𝑈 = (PCl‘𝐾) |
Ref | Expression |
---|---|
pclfvalN | ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) | |
2 | pclfval.c | . . 3 ⊢ 𝑈 = (PCl‘𝐾) | |
3 | fveq2 6103 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) | |
4 | pclfval.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) |
6 | 5 | pweqd 4113 | . . . . 5 ⊢ (𝑘 = 𝐾 → 𝒫 (Atoms‘𝑘) = 𝒫 𝐴) |
7 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾)) | |
8 | pclfval.s | . . . . . . . 8 ⊢ 𝑆 = (PSubSp‘𝐾) | |
9 | 7, 8 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆) |
10 | 9 | rabeqdv 3167 | . . . . . 6 ⊢ (𝑘 = 𝐾 → {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) |
11 | 10 | inteqd 4415 | . . . . 5 ⊢ (𝑘 = 𝐾 → ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) |
12 | 6, 11 | mpteq12dv 4663 | . . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})) |
13 | df-pclN 34192 | . . . 4 ⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦})) | |
14 | fvex 6113 | . . . . . . 7 ⊢ (Atoms‘𝐾) ∈ V | |
15 | 4, 14 | eqeltri 2684 | . . . . . 6 ⊢ 𝐴 ∈ V |
16 | 15 | pwex 4774 | . . . . 5 ⊢ 𝒫 𝐴 ∈ V |
17 | 16 | mptex 6390 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) ∈ V |
18 | 12, 13, 17 | fvmpt 6191 | . . 3 ⊢ (𝐾 ∈ V → (PCl‘𝐾) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})) |
19 | 2, 18 | syl5eq 2656 | . 2 ⊢ (𝐾 ∈ V → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∩ cint 4410 ↦ cmpt 4643 ‘cfv 5804 Atomscatm 33568 PSubSpcpsubsp 33800 PClcpclN 34191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-pclN 34192 |
This theorem is referenced by: pclvalN 34194 |
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