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Theorem psubclsetN 36523
Description: The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclset.a 𝐴 = (Atoms‘𝐾)
psubclset.p = (⊥𝑃𝐾)
psubclset.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubclsetN (𝐾𝐵𝐶 = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
Distinct variable groups:   𝐴,𝑠   𝐾,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝐶(𝑠)   (𝑠)

Proof of Theorem psubclsetN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3433 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubclset.c . . 3 𝐶 = (PSubCl‘𝐾)
3 fveq2 6499 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubclset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2832 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 3889 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6499 . . . . . . . . 9 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
8 psubclset.p . . . . . . . . 9 = (⊥𝑃𝐾)
97, 8syl6eqr 2832 . . . . . . . 8 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
109fveq1d 6501 . . . . . . . 8 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘𝑠) = ( 𝑠))
119, 10fveq12d 6506 . . . . . . 7 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = ( ‘( 𝑠)))
1211eqeq1d 2780 . . . . . 6 (𝑘 = 𝐾 → (((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠 ↔ ( ‘( 𝑠)) = 𝑠))
136, 12anbi12d 621 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠) ↔ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)))
1413abbidv 2843 . . . 4 (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
15 df-psubclN 36522 . . . 4 PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)})
164fvexi 6513 . . . . . 6 𝐴 ∈ V
1716pwex 5134 . . . . 5 𝒫 𝐴 ∈ V
18 selpw 4429 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
1918anbi1i 614 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠) ↔ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠))
2019abbii 2844 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)}
21 ssab2 3945 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
2220, 21eqsstr3i 3892 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
2317, 22ssexi 5082 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ∈ V
2414, 15, 23fvmpt 6595 . . 3 (𝐾 ∈ V → (PSubCl‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
252, 24syl5eq 2826 . 2 (𝐾 ∈ V → 𝐶 = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
261, 25syl 17 1 (𝐾𝐵𝐶 = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  {cab 2758  Vcvv 3415  wss 3829  𝒫 cpw 4422  cfv 6188  Atomscatm 35850  𝑃cpolN 36489  PSubClcpscN 36521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-psubclN 36522
This theorem is referenced by:  ispsubclN  36524
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