Theorem elpcliN 39850

Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elpcli.s	⊢ 𝑆 = (PSubSp‘𝐾)
elpcli.c	⊢ 𝑈 = (PCl‘𝐾)

Assertion

Ref	Expression
elpcliN	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) ∧ 𝑄 ∈ (𝑈‘𝑋)) → 𝑄 ∈ 𝑌)

Proof of Theorem elpcliN

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝐾 ∈ 𝑉)
2		simp2 1137	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ 𝑌)
3		eqid 2740	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		elpcli.s	. . . . . . . . 9 ⊢ 𝑆 = (PSubSp‘𝐾)
5	3, 4	psubssat 39711	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
6	5	3adant2 1131	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
7	2, 6	sstrd 4019	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8		elpcli.c	. . . . . . 7 ⊢ 𝑈 = (PCl‘𝐾)
9	3, 4, 8	pclvalN 39847	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑈‘𝑋) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧})
10	1, 7, 9	syl2anc 583	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑈‘𝑋) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧})
11	10	eleq2d 2830	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧}))
12		elintrabg 4985	. . . . 5 ⊢ (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} → (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} ↔ ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧)))
13	12	ibi 267	. . . 4 ⊢ (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} → ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧))
14	11, 13	biimtrdi 253	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) → ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧)))
15		sseq2 4035	. . . . . . . 8 ⊢ (𝑧 = 𝑌 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑌))
16		eleq2 2833	. . . . . . . 8 ⊢ (𝑧 = 𝑌 → (𝑄 ∈ 𝑧 ↔ 𝑄 ∈ 𝑌))
17	15, 16	imbi12d 344	. . . . . . 7 ⊢ (𝑧 = 𝑌 → ((𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) ↔ (𝑋 ⊆ 𝑌 → 𝑄 ∈ 𝑌)))
18	17	rspccv 3632	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → (𝑌 ∈ 𝑆 → (𝑋 ⊆ 𝑌 → 𝑄 ∈ 𝑌)))
19	18	com13 88	. . . . 5 ⊢ (𝑋 ⊆ 𝑌 → (𝑌 ∈ 𝑆 → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌)))
20	19	imp 406	. . . 4 ⊢ ((𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌))
21	20	3adant1 1130	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌))
22	14, 21	syld 47	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) → 𝑄 ∈ 𝑌))
23	22	imp 406	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) ∧ 𝑄 ∈ (𝑈‘𝑋)) → 𝑄 ∈ 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443   ⊆ wss 3976  ∩ cint 4970  ‘cfv 6573  Atomscatm 39219  PSubSpcpsubsp 39453  PClcpclN 39844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-psubsp 39460  df-pclN 39845

This theorem is referenced by:  pclfinclN  39907