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Theorem psubspset 39727
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 3499 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubspset.s . . 3 𝑆 = (PSubSp‘𝐾)
3 fveq2 6907 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubspset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2793 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 4028 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6907 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
8 psubspset.j . . . . . . . . . . . . 13 = (join‘𝐾)
97, 8eqtr4di 2793 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
109oveqd 7448 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 𝑞))
1110breq2d 5160 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 𝑞)))
12 fveq2 6907 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
13 psubspset.l . . . . . . . . . . . 12 = (le‘𝐾)
1412, 13eqtr4di 2793 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
1514breqd 5159 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝 𝑞) ↔ 𝑟 (𝑝 𝑞)))
1611, 15bitrd 279 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟 (𝑝 𝑞)))
1716imbi1d 341 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
185, 17raleqbidv 3344 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
19182ralbidv 3219 . . . . . 6 (𝑘 = 𝐾 → (∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
206, 19anbi12d 632 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))))
2120abbidv 2806 . . . 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 39486 . . . 4 PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠))})
234fvexi 6921 . . . . . 6 𝐴 ∈ V
2423pwex 5386 . . . . 5 𝒫 𝐴 ∈ V
25 velpw 4610 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
2625anbi1i 624 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
2726abbii 2807 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))}
28 ssab2 4089 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
2927, 28eqsstrri 4031 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
3024, 29ssexi 5328 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ∈ V
3121, 22, 30fvmpt 7016 . . 3 (𝐾 ∈ V → (PSubSp‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
322, 31eqtrid 2787 . 2 (𝐾 ∈ V → 𝑆 = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
331, 32syl 17 1 (𝐾𝐵𝑆 = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  Atomscatm 39245  PSubSpcpsubsp 39479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-psubsp 39486
This theorem is referenced by:  ispsubsp  39728
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