Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > pclssN | Structured version Visualization version GIF version |
Description: Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclss.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pclss.c | ⊢ 𝑈 = (PCl‘𝐾) |
Ref | Expression |
---|---|
pclssN | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3928 | . . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦)) | |
2 | 1 | 3ad2ant2 1133 | . . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦)) |
3 | 2 | adantr 481 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑦 ∈ (PSubSp‘𝐾)) → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦)) |
4 | 3 | ss2rabdv 4009 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦}) |
5 | intss 4900 | . . 3 ⊢ ({𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦} → ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦} ⊆ ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦}) | |
6 | 4, 5 | syl 17 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦} ⊆ ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦}) |
7 | simp1 1135 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝐾 ∈ 𝑉) | |
8 | sstr 3929 | . . . 4 ⊢ ((𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴) | |
9 | 8 | 3adant1 1129 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴) |
10 | pclss.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | eqid 2738 | . . . 4 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾) | |
12 | pclss.c | . . . 4 ⊢ 𝑈 = (PCl‘𝐾) | |
13 | 10, 11, 12 | pclvalN 37904 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦}) |
14 | 7, 9, 13 | syl2anc 584 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦}) |
15 | 10, 11, 12 | pclvalN 37904 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑌) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦}) |
16 | 15 | 3adant2 1130 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑌) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦}) |
17 | 6, 14, 16 | 3sstr4d 3968 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 ∩ cint 4879 ‘cfv 6433 Atomscatm 37277 PSubSpcpsubsp 37510 PClcpclN 37901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-psubsp 37517 df-pclN 37902 |
This theorem is referenced by: pclbtwnN 37911 pclunN 37912 pclfinN 37914 pclss2polN 37935 pclfinclN 37964 |
Copyright terms: Public domain | W3C validator |