Theorem pclssN 40013

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 3937	. . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦))
2	1	3ad2ant2 1134	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦))
3	2	adantr 480	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑦 ∈ (PSubSp‘𝐾)) → (𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦))
4	3	ss2rabdv 4024	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦})
5		intss 4919	. . 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 1136	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝐾 ∈ 𝑉)
8		sstr 3939	. . . 4 ⊢ ((𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
9	8	3adant1 1130	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
10		pclss.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
11		eqid 2733	. . . 4 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
12		pclss.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
13	10, 11, 12	pclvalN 40009	. . 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 40009	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑌) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦})
16	15	3adant2 1131	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑌) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦})
17	6, 14, 16	3sstr4d 3986	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘𝑌))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {crab 3396   ⊆ wss 3898  ∩ cint 4897  ‘cfv 6486  Atomscatm 39382  PSubSpcpsubsp 39615  PClcpclN 40006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-psubsp 39622  df-pclN 40007

This theorem is referenced by:  pclbtwnN  40016  pclunN  40017  pclfinN  40019  pclss2polN  40040  pclfinclN  40069