Theorem elpcliN 40149

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 2736	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		elpcli.s	. . . . . . . . 9 ⊢ 𝑆 = (PSubSp‘𝐾)
5	3, 4	psubssat 40010	. . . . . . . 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 3944	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8		elpcli.c	. . . . . . 7 ⊢ 𝑈 = (PCl‘𝐾)
9	3, 4, 8	pclvalN 40146	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑈‘𝑋) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧})
10	1, 7, 9	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑈‘𝑋) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧})
11	10	eleq2d 2822	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧}))
12		elintrabg 4916	. . . . 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 3960	. . . . . . . 8 ⊢ (𝑧 = 𝑌 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑌))
16		eleq2 2825	. . . . . . . 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 3573	. . . . . 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 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399   ⊆ wss 3901  ∩ cint 4902  ‘cfv 6492  Atomscatm 39519  PSubSpcpsubsp 39752  PClcpclN 40143

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-psubsp 39759  df-pclN 40144

This theorem is referenced by:  pclfinclN  40206