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Theorem psubclsetN 37225
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 3462 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubclset.c . . 3 𝐶 = (PSubCl‘𝐾)
3 fveq2 6649 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubclset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2854 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 3950 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6649 . . . . . . . . 9 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
8 psubclset.p . . . . . . . . 9 = (⊥𝑃𝐾)
97, 8eqtr4di 2854 . . . . . . . 8 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
109fveq1d 6651 . . . . . . . 8 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘𝑠) = ( 𝑠))
119, 10fveq12d 6656 . . . . . . 7 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = ( ‘( 𝑠)))
1211eqeq1d 2803 . . . . . 6 (𝑘 = 𝐾 → (((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠 ↔ ( ‘( 𝑠)) = 𝑠))
136, 12anbi12d 633 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠) ↔ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)))
1413abbidv 2865 . . . 4 (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
15 df-psubclN 37224 . . . 4 PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)})
164fvexi 6663 . . . . . 6 𝐴 ∈ V
1716pwex 5249 . . . . 5 𝒫 𝐴 ∈ V
18 velpw 4505 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
1918anbi1i 626 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠) ↔ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠))
2019abbii 2866 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)}
21 ssab2 4009 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
2220, 21eqsstrri 3953 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
2317, 22ssexi 5193 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ∈ V
2414, 15, 23fvmpt 6749 . . 3 (𝐾 ∈ V → (PSubCl‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
252, 24syl5eq 2848 . 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 399   = wceq 1538  wcel 2112  {cab 2779  Vcvv 3444  wss 3884  𝒫 cpw 4500  cfv 6328  Atomscatm 36552  𝑃cpolN 37191  PSubClcpscN 37223
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-psubclN 37224
This theorem is referenced by:  ispsubclN  37226
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