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Theorem psubspset 36940
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 3497 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubspset.s . . 3 𝑆 = (PSubSp‘𝐾)
3 fveq2 6651 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubspset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2877 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 3983 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6651 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
8 psubspset.j . . . . . . . . . . . . 13 = (join‘𝐾)
97, 8syl6eqr 2877 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
109oveqd 7155 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 𝑞))
1110breq2d 5059 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 𝑞)))
12 fveq2 6651 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
13 psubspset.l . . . . . . . . . . . 12 = (le‘𝐾)
1412, 13syl6eqr 2877 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
1514breqd 5058 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝 𝑞) ↔ 𝑟 (𝑝 𝑞)))
1611, 15bitrd 282 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟 (𝑝 𝑞)))
1716imbi1d 345 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
185, 17raleqbidv 3392 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
19182ralbidv 3193 . . . . . 6 (𝑘 = 𝐾 → (∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
206, 19anbi12d 633 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))))
2120abbidv 2888 . . . 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 36699 . . . 4 PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠))})
234fvexi 6665 . . . . . 6 𝐴 ∈ V
2423pwex 5262 . . . . 5 𝒫 𝐴 ∈ V
25 velpw 4525 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
2625anbi1i 626 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
2726abbii 2889 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))}
28 ssab2 4039 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
2927, 28eqsstrri 3986 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
3024, 29ssexi 5207 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ∈ V
3121, 22, 30fvmpt 6749 . . 3 (𝐾 ∈ V → (PSubSp‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
322, 31syl5eq 2871 . 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 399   = wceq 1538  wcel 2115  {cab 2802  wral 3132  Vcvv 3479  wss 3918  𝒫 cpw 4520   class class class wbr 5047  cfv 6336  (class class class)co 7138  lecple 16561  joincjn 17543  Atomscatm 36459  PSubSpcpsubsp 36692
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7141  df-psubsp 36699
This theorem is referenced by:  ispsubsp  36941
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