Theorem pclbtwnN 40343

Description: A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem pclbtwnN

Step	Hyp	Ref	Expression
1		simprr 773	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑋 ⊆ (𝑈‘𝑌))
2		simpll 767	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝐾 ∈ 𝑉)
3		simprl 771	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑌 ⊆ 𝑋)
4		eqid 2736	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		pclid.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
6	4, 5	psubssat 40200	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
7	6	adantr 480	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑋 ⊆ (Atoms‘𝐾))
8		pclid.c	. . . . 5 ⊢ 𝑈 = (PCl‘𝐾)
9	4, 8	pclssN 40340	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑈‘𝑌) ⊆ (𝑈‘𝑋))
10	2, 3, 7, 9	syl3anc 1374	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → (𝑈‘𝑌) ⊆ (𝑈‘𝑋))
11	5, 8	pclidN 40342	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋)
12	11	adantr 480	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → (𝑈‘𝑋) = 𝑋)
13	10, 12	sseqtrd 3958	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → (𝑈‘𝑌) ⊆ 𝑋)
14	1, 13	eqssd 3939	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑋 = (𝑈‘𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ‘cfv 6498  Atomscatm 39709  PSubSpcpsubsp 39942  PClcpclN 40333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-psubsp 39949  df-pclN 40334

This theorem is referenced by:  pclfinN  40346