Theorem psubssat 38620

Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)

Hypotheses

Ref	Expression
atpsub.a	⊢ 𝐴 = (Atoms‘𝐾)
atpsub.s	⊢ 𝑆 = (PSubSp‘𝐾)

Assertion

Ref	Expression
psubssat	⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴)

Proof of Theorem psubssat

Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
2		eqid 2732	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		atpsub.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		atpsub.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
5	1, 2, 3, 4	ispsubsp 38611	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋))))
6	5	simprbda 499	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3948   class class class wbr 5148  ‘cfv 6543  (class class class)co 7408  lecple 17203  joincjn 18263  Atomscatm 38128  PSubSpcpsubsp 38362

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-psubsp 38369

This theorem is referenced by:  psubatN  38621  paddidm  38707  paddclN  38708  paddss  38711  pmodlem1  38712  pmod1i  38714  pmodl42N  38717  elpcliN  38759  pclidN  38762  pclbtwnN  38763  pclunN  38764  pclun2N  38765  pclfinN  38766  polssatN  38774  psubclsubN  38806