Theorem pclssN 41154

Description: Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclss.a	⊢ 𝐴 = (Atoms‘𝐾)
pclss.c	⊢ 𝑈 = (PCl‘𝐾)

Assertion

Ref	Expression
pclssN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘𝑌))

Proof of Theorem pclssN

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3920	. . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦))
2	1	3ad2ant2 1152	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦))
3	2	adantr 486	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑦 ∈ (PSubSp‘𝐾)) → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦))
4	3	ss2rabdv 4003	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦})
5		intss 4932	. . 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 18	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦} ⊆ ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦})
7		simp1 1154	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝐾 ∈ 𝑉)
8		sstr 3921	. . . 4 ⊢ ((𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
9	8	3adant1 1148	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
10		pclss.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
11		eqid 2763	. . . 4 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
12		pclss.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
13	10, 11, 12	pclvalN 41150	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦})
14	7, 9, 13	syl2anc 596	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦})
15	10, 11, 12	pclvalN 41150	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑌) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦})
16	15	3adant2 1149	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {crab 3416   ⊆ wss 3900  ∩ cint 4910  ‘cfv 6561  Atomscatm 40523  PSubSpcpsubsp 40756  PClcpclN 41147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-psubsp 40763  df-pclN 41148

This theorem is used by:  pclbtwnN  41157  pclunN  41158  pclfinN  41160  pclss2polN  41181  pclfinclN  41210