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Theorem psubclsetN 39110
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 = (Atomsβ€˜πΎ)
psubclset.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
psubclset.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
psubclsetN (𝐾 ∈ 𝐡 β†’ 𝐢 = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)})
Distinct variable groups:   𝐴,𝑠   𝐾,𝑠
Allowed substitution hints:   𝐡(𝑠)   𝐢(𝑠)   βŠ₯ (𝑠)

Proof of Theorem psubclsetN
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3491 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 psubclset.c . . 3 𝐢 = (PSubClβ€˜πΎ)
3 fveq2 6890 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 psubclset.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2788 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
65sseq2d 4013 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑠 βŠ† (Atomsβ€˜π‘˜) ↔ 𝑠 βŠ† 𝐴))
7 fveq2 6890 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (βŠ₯π‘ƒβ€˜π‘˜) = (βŠ₯π‘ƒβ€˜πΎ))
8 psubclset.p . . . . . . . . 9 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
97, 8eqtr4di 2788 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (βŠ₯π‘ƒβ€˜π‘˜) = βŠ₯ )
109fveq1d 6892 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ ) = ( βŠ₯ β€˜π‘ ))
119, 10fveq12d 6897 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ )) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )))
1211eqeq1d 2732 . . . . . 6 (π‘˜ = 𝐾 β†’ (((βŠ₯π‘ƒβ€˜π‘˜)β€˜((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ )) = 𝑠 ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠))
136, 12anbi12d 629 . . . . 5 (π‘˜ = 𝐾 β†’ ((𝑠 βŠ† (Atomsβ€˜π‘˜) ∧ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ )) = 𝑠) ↔ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)))
1413abbidv 2799 . . . 4 (π‘˜ = 𝐾 β†’ {𝑠 ∣ (𝑠 βŠ† (Atomsβ€˜π‘˜) ∧ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ )) = 𝑠)} = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)})
15 df-psubclN 39109 . . . 4 PSubCl = (π‘˜ ∈ V ↦ {𝑠 ∣ (𝑠 βŠ† (Atomsβ€˜π‘˜) ∧ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ )) = 𝑠)})
164fvexi 6904 . . . . . 6 𝐴 ∈ V
1716pwex 5377 . . . . 5 𝒫 𝐴 ∈ V
18 velpw 4606 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 βŠ† 𝐴)
1918anbi1i 622 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠) ↔ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠))
2019abbii 2800 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)} = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)}
21 ssab2 4075 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)} βŠ† 𝒫 𝐴
2220, 21eqsstrri 4016 . . . . 5 {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)} βŠ† 𝒫 𝐴
2317, 22ssexi 5321 . . . 4 {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)} ∈ V
2414, 15, 23fvmpt 6997 . . 3 (𝐾 ∈ V β†’ (PSubClβ€˜πΎ) = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)})
252, 24eqtrid 2782 . 2 (𝐾 ∈ V β†’ 𝐢 = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)})
261, 25syl 17 1 (𝐾 ∈ 𝐡 β†’ 𝐢 = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  Vcvv 3472   βŠ† wss 3947  π’« cpw 4601  β€˜cfv 6542  Atomscatm 38436  βŠ₯𝑃cpolN 39076  PSubClcpscN 39108
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-psubclN 39109
This theorem is referenced by:  ispsubclN  39111
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