Theorem pclssN 41045

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 3915	. . . . . 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 4000	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌 ⊆ 𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦})
5		intss 4928	. . 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 3916	. . . 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 2760	. . . 4 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
12		pclss.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
13	10, 11, 12	pclvalN 41041	. . 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 41041	. . 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 3963	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3895  ∩ cint 4906  ‘cfv 6539  Atomscatm 40414  PSubSpcpsubsp 40647  PClcpclN 41038

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-psubsp 40654  df-pclN 41039

This theorem is used by:  pclbtwnN  41048  pclunN  41049  pclfinN  41051  pclss2polN  41072  pclfinclN  41101