Theorem pclbtwnN 40360

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 2737	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		pclid.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
6	4, 5	psubssat 40217	. . . . 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 40357	. . . 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 40359	. . . 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 3959	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → (𝑈‘𝑌) ⊆ 𝑋)
14	1, 13	eqssd 3940	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑋 = (𝑈‘𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ‘cfv 6493  Atomscatm 39726  PSubSpcpsubsp 39959  PClcpclN 40350

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-psubsp 39966  df-pclN 40351

This theorem is referenced by:  pclfinN  40363