Theorem elpcliN 40918

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 2761	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		elpcli.s	. . . . . . . . 9 ⊢ 𝑆 = (PSubSp‘𝐾)
5	3, 4	psubssat 40779	. . . . . . . 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 3941	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8		elpcli.c	. . . . . . 7 ⊢ 𝑈 = (PCl‘𝐾)
9	3, 4, 8	pclvalN 40915	. . . . . 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 2847	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧}))
12		elintrabg 4921	. . . . 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 3957	. . . . . . . 8 ⊢ (𝑧 = 𝑌 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑌))
16		eleq2 2850	. . . . . . . 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 3574	. . . . . 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 2145  ∀wral 3077  {crab 3413   ⊆ wss 3899  ∩ cint 4907  ‘cfv 6531  Atomscatm 40288  PSubSpcpsubsp 40521  PClcpclN 40912

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-psubsp 40528  df-pclN 40913

This theorem is used by:  pclfinclN  40975