Theorem elpcliN 40707

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 1154	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝐾 ∈ 𝑉)
2		simp2 1155	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ 𝑌)
3		eqid 2766	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		elpcli.s	. . . . . . . . 9 ⊢ 𝑆 = (PSubSp‘𝐾)
5	3, 4	psubssat 40568	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
6	5	3adant2 1149	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
7	2, 6	sstrd 3950	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8		elpcli.c	. . . . . . 7 ⊢ 𝑈 = (PCl‘𝐾)
9	3, 4, 8	pclvalN 40704	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑈‘𝑋) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧})
10	1, 7, 9	syl2anc 596	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑈‘𝑋) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧})
11	10	eleq2d 2852	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧}))
12		elintrabg 4931	. . . . 5 ⊢ (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} → (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} ↔ ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧)))
13	12	ibi 270	. . . 4 ⊢ (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧} → ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧))
14	11, 13	biimtrdi 256	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) → ∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧)))
15		sseq2 3966	. . . . . . . 8 ⊢ (𝑧 = 𝑌 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑌))
16		eleq2 2855	. . . . . . . 8 ⊢ (𝑧 = 𝑌 → (𝑄 ∈ 𝑧 ↔ 𝑄 ∈ 𝑌))
17	15, 16	imbi12d 347	. . . . . . 7 ⊢ (𝑧 = 𝑌 → ((𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) ↔ (𝑋 ⊆ 𝑌 → 𝑄 ∈ 𝑌)))
18	17	rspccv 3581	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → (𝑌 ∈ 𝑆 → (𝑋 ⊆ 𝑌 → 𝑄 ∈ 𝑌)))
19	18	com13 89	. . . . 5 ⊢ (𝑋 ⊆ 𝑌 → (𝑌 ∈ 𝑆 → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌)))
20	19	imp 412	. . . 4 ⊢ ((𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌))
21	20	3adant1 1148	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (∀𝑧 ∈ 𝑆 (𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧) → 𝑄 ∈ 𝑌))
22	14, 21	syld 48	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) → 𝑄 ∈ 𝑌))
23	22	imp 412	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) ∧ 𝑄 ∈ (𝑈‘𝑋)) → 𝑄 ∈ 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  {crab 3419   ⊆ wss 3908  ∩ cint 4917  ‘cfv 6543  Atomscatm 40077  PSubSpcpsubsp 40310  PClcpclN 40701

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 40317  df-pclN 40702

This theorem is used by:  pclfinclN  40764