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Theorem psubspset 36760
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 3510 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubspset.s . . 3 𝑆 = (PSubSp‘𝐾)
3 fveq2 6663 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubspset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2871 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 3996 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6663 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
8 psubspset.j . . . . . . . . . . . . 13 = (join‘𝐾)
97, 8syl6eqr 2871 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
109oveqd 7162 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 𝑞))
1110breq2d 5069 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 𝑞)))
12 fveq2 6663 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
13 psubspset.l . . . . . . . . . . . 12 = (le‘𝐾)
1412, 13syl6eqr 2871 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
1514breqd 5068 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝 𝑞) ↔ 𝑟 (𝑝 𝑞)))
1611, 15bitrd 280 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟 (𝑝 𝑞)))
1716imbi1d 343 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
185, 17raleqbidv 3399 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
19182ralbidv 3196 . . . . . 6 (𝑘 = 𝐾 → (∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠) ↔ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
206, 19anbi12d 630 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))))
2120abbidv 2882 . . . 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 36519 . . . 4 PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝𝑠𝑞𝑠𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟𝑠))})
234fvexi 6677 . . . . . 6 𝐴 ∈ V
2423pwex 5272 . . . . 5 𝒫 𝐴 ∈ V
25 velpw 4543 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
2625anbi1i 623 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)) ↔ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠)))
2726abbii 2883 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))}
28 ssab2 4052 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
2927, 28eqsstrri 3999 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ⊆ 𝒫 𝐴
3024, 29ssexi 5217 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))} ∈ V
3121, 22, 30fvmpt 6761 . . 3 (𝐾 ∈ V → (PSubSp‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})
322, 31syl5eq 2865 . 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 396   = wceq 1528  wcel 2105  {cab 2796  wral 3135  Vcvv 3492  wss 3933  𝒫 cpw 4535   class class class wbr 5057  cfv 6348  (class class class)co 7145  lecple 16560  joincjn 17542  Atomscatm 36279  PSubSpcpsubsp 36512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-psubsp 36519
This theorem is referenced by:  ispsubsp  36761
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