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Theorem psubspset 34496
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 3203 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubspset.s . . 3 𝑆 = (PSubSp‘𝐾)
3 fveq2 6150 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubspset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2678 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 3617 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6150 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
8 psubspset.j . . . . . . . . . . . . 13 = (join‘𝐾)
97, 8syl6eqr 2678 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
109oveqd 6622 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 𝑞))
1110breq2d 4630 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 𝑞)))
12 fveq2 6150 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
13 psubspset.l . . . . . . . . . . . 12 = (le‘𝐾)
1412, 13syl6eqr 2678 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
1514breqd 4629 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝 𝑞) ↔ 𝑟 (𝑝 𝑞)))
1611, 15bitrd 268 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟 (𝑝 𝑞)))
1716imbi1d 331 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
185, 17raleqbidv 3146 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
19182ralbidv 2988 . . . . . 6 (𝑘 = 𝐾 → (∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
206, 19anbi12d 746 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))))
2120abbidv 2744 . . . 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 34255 . . . 4 PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠))})
23 fvex 6160 . . . . . . 7 (Atoms‘𝐾) ∈ V
244, 23eqeltri 2700 . . . . . 6 𝐴 ∈ V
2524pwex 4813 . . . . 5 𝒫 𝐴 ∈ V
26 selpw 4142 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
2726anbi1i 730 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
2827abbii 2742 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))}
29 ssab2 3670 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
3028, 29eqsstr3i 3620 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
3125, 30ssexi 4768 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ∈ V
3221, 22, 31fvmpt 6240 . . 3 (𝐾 ∈ V → (PSubSp‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
332, 32syl5eq 2672 . 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 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  Vcvv 3191  wss 3560  𝒫 cpw 4135   class class class wbr 4618  cfv 5850  (class class class)co 6605  lecple 15864  joincjn 16860  Atomscatm 34016  PSubSpcpsubsp 34248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-psubsp 34255
This theorem is referenced by:  ispsubsp  34497
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