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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pclvalN | Structured version Visualization version GIF version |
Description: Value of 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 |
---|---|
pclvalN | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pclfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | 1 | fvexi 6892 | . . 3 ⊢ 𝐴 ∈ V |
3 | 2 | elpw2 5338 | . 2 ⊢ (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴) |
4 | pclfval.s | . . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾) | |
5 | pclfval.c | . . . . . 6 ⊢ 𝑈 = (PCl‘𝐾) | |
6 | 1, 4, 5 | pclfvalN 38565 | . . . . 5 ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})) |
7 | 6 | fveq1d 6880 | . . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝑈‘𝑋) = ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋)) |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑈‘𝑋) = ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋)) |
9 | eqid 2731 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) | |
10 | sseq1 4003 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑦)) | |
11 | 10 | rabbidv 3439 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
12 | 11 | inteqd 4948 | . . . 4 ⊢ (𝑥 = 𝑋 → ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
13 | simpr 485 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴) | |
14 | elpwi 4603 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝒫 𝐴 → 𝑋 ⊆ 𝐴) | |
15 | 14 | adantl 482 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ⊆ 𝐴) |
16 | 1, 4 | atpsubN 38429 | . . . . . . . . 9 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
17 | 16 | adantr 481 | . . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ 𝑆) |
18 | sseq2 4004 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴)) | |
19 | 18 | elrab3 3680 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 → (𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ 𝑋 ⊆ 𝐴)) |
20 | 17, 19 | syl 17 | . . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ 𝑋 ⊆ 𝐴)) |
21 | 15, 20 | mpbird 256 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
22 | 21 | ne0d 4331 | . . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ≠ ∅) |
23 | intex 5330 | . . . . 5 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ≠ ∅ ↔ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ V) | |
24 | 22, 23 | sylib 217 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ V) |
25 | 9, 12, 13, 24 | fvmptd3 7007 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
26 | 8, 25 | eqtrd 2771 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
27 | 3, 26 | sylan2br 595 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 {crab 3431 Vcvv 3473 ⊆ wss 3944 ∅c0 4318 𝒫 cpw 4596 ∩ cint 4943 ↦ cmpt 5224 ‘cfv 6532 Atomscatm 37938 PSubSpcpsubsp 38172 PClcpclN 38563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-psubsp 38179 df-pclN 38564 |
This theorem is referenced by: pclclN 38567 elpclN 38568 elpcliN 38569 pclssN 38570 pclssidN 38571 pclidN 38572 |
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