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Theorem psubclsetN 39938
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 3501 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubclset.c . . 3 𝐶 = (PSubCl‘𝐾)
3 fveq2 6906 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubclset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2795 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 4016 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6906 . . . . . . . . 9 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
8 psubclset.p . . . . . . . . 9 = (⊥𝑃𝐾)
97, 8eqtr4di 2795 . . . . . . . 8 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
109fveq1d 6908 . . . . . . . 8 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘𝑠) = ( 𝑠))
119, 10fveq12d 6913 . . . . . . 7 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = ( ‘( 𝑠)))
1211eqeq1d 2739 . . . . . 6 (𝑘 = 𝐾 → (((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠 ↔ ( ‘( 𝑠)) = 𝑠))
136, 12anbi12d 632 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠) ↔ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)))
1413abbidv 2808 . . . 4 (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
15 df-psubclN 39937 . . . 4 PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)})
164fvexi 6920 . . . . . 6 𝐴 ∈ V
1716pwex 5380 . . . . 5 𝒫 𝐴 ∈ V
18 velpw 4605 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
1918anbi1i 624 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠) ↔ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠))
2019abbii 2809 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)}
21 ssab2 4079 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
2220, 21eqsstrri 4031 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
2317, 22ssexi 5322 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ∈ V
2414, 15, 23fvmpt 7016 . . 3 (𝐾 ∈ V → (PSubCl‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
252, 24eqtrid 2789 . 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 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  wss 3951  𝒫 cpw 4600  cfv 6561  Atomscatm 39264  𝑃cpolN 39904  PSubClcpscN 39936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-psubclN 39937
This theorem is referenced by:  ispsubclN  39939
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