Theorem pclbtwnN 40389

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 778	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑋 ⊆ (𝑈‘𝑌))
2		simpll 772	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝐾 ∈ 𝑉)
3		simprl 776	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑌 ⊆ 𝑋)
4		eqid 2739	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		pclid.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
6	4, 5	psubssat 40246	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
7	6	adantr 481	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑋 ⊆ (Atoms‘𝐾))
8		pclid.c	. . . . 5 ⊢ 𝑈 = (PCl‘𝐾)
9	4, 8	pclssN 40386	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑈‘𝑌) ⊆ (𝑈‘𝑋))
10	2, 3, 7, 9	syl3anc 1379	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → (𝑈‘𝑌) ⊆ (𝑈‘𝑋))
11	5, 8	pclidN 40388	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋)
12	11	adantr 481	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → (𝑈‘𝑋) = 𝑋)
13	10, 12	sseqtrd 3951	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → (𝑈‘𝑌) ⊆ 𝑋)
14	1, 13	eqssd 3932	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑋 = (𝑈‘𝑌))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  ‘cfv 6485  Atomscatm 39755  PSubSpcpsubsp 39988  PClcpclN 40379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-psubsp 39995  df-pclN 40380

This theorem is referenced by:  pclfinN  40392