Theorem pclssN 40709

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 3947	. . . . . 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 4032	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦})
5		intss 4939	. . 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 3948	. . . 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 2766	. . . 4 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
12		pclss.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
13	10, 11, 12	pclvalN 40705	. . 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 40705	. . 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 3995	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {crab 3419   ⊆ wss 3908  ∩ cint 4917  ‘cfv 6543  Atomscatm 40078  PSubSpcpsubsp 40311  PClcpclN 40702

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-psubsp 40318  df-pclN 40703

This theorem is used by:  pclbtwnN  40712  pclunN  40713  pclfinN  40715  pclss2polN  40736  pclfinclN  40765