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Theorem psubspset 34048
 Description: The set of projective subspaces in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
Hypotheses
Ref Expression
psubspset.l = (le‘𝐾)
psubspset.j = (join‘𝐾)
psubspset.a 𝐴 = (Atoms‘𝐾)
psubspset.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
psubspset (𝐾𝐵𝑆 = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
Distinct variable groups:   𝑠,𝑟,𝐴   𝑞,𝑝,𝑟,𝑠,𝐾
Allowed substitution hints:   𝐴(𝑞,𝑝)   𝐵(𝑠,𝑟,𝑞,𝑝)   𝑆(𝑠,𝑟,𝑞,𝑝)   (𝑠,𝑟,𝑞,𝑝)   (𝑠,𝑟,𝑞,𝑝)

Proof of Theorem psubspset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubspset.s . . 3 𝑆 = (PSubSp‘𝐾)
3 fveq2 6103 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubspset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2662 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 3596 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6103 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
8 psubspset.j . . . . . . . . . . . . 13 = (join‘𝐾)
97, 8syl6eqr 2662 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
109oveqd 6566 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 𝑞))
1110breq2d 4595 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 𝑞)))
12 fveq2 6103 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
13 psubspset.l . . . . . . . . . . . 12 = (le‘𝐾)
1412, 13syl6eqr 2662 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
1514breqd 4594 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝 𝑞) ↔ 𝑟 (𝑝 𝑞)))
1611, 15bitrd 267 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟 (𝑝 𝑞)))
1716imbi1d 330 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
185, 17raleqbidv 3129 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
19182ralbidv 2972 . . . . . 6 (𝑘 = 𝐾 → (∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
206, 19anbi12d 743 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))))
2120abbidv 2728 . . . 4 (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠))} = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
22 df-psubsp 33807 . . . 4 PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠))})
23 fvex 6113 . . . . . . 7 (Atoms‘𝐾) ∈ V
244, 23eqeltri 2684 . . . . . 6 𝐴 ∈ V
2524pwex 4774 . . . . 5 𝒫 𝐴 ∈ V
26 selpw 4115 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
2726anbi1i 727 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
2827abbii 2726 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))}
29 ssab2 3649 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
3028, 29eqsstr3i 3599 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
3125, 30ssexi 4731 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ∈ V
3221, 22, 31fvmpt 6191 . . 3 (𝐾 ∈ V → (PSubSp‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
332, 32syl5eq 2656 . 2 (𝐾 ∈ V → 𝑆 = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
341, 33syl 17 1 (𝐾𝐵𝑆 = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  Atomscatm 33568  PSubSpcpsubsp 33800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-psubsp 33807 This theorem is referenced by:  ispsubsp  34049
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