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Theorem psubclsetN 38402
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 3464 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 psubclset.c . . 3 𝐢 = (PSubClβ€˜πΎ)
3 fveq2 6843 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 psubclset.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2795 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
65sseq2d 3977 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑠 βŠ† (Atomsβ€˜π‘˜) ↔ 𝑠 βŠ† 𝐴))
7 fveq2 6843 . . . . . . . . 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 2795 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (βŠ₯π‘ƒβ€˜π‘˜) = βŠ₯ )
109fveq1d 6845 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ ) = ( βŠ₯ β€˜π‘ ))
119, 10fveq12d 6850 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ )) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )))
1211eqeq1d 2739 . . . . . 6 (π‘˜ = 𝐾 β†’ (((βŠ₯π‘ƒβ€˜π‘˜)β€˜((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ )) = 𝑠 ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠))
136, 12anbi12d 632 . . . . 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 2806 . . . 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 38401 . . . 4 PSubCl = (π‘˜ ∈ V ↦ {𝑠 ∣ (𝑠 βŠ† (Atomsβ€˜π‘˜) ∧ ((βŠ₯π‘ƒβ€˜π‘˜)β€˜((βŠ₯π‘ƒβ€˜π‘˜)β€˜π‘ )) = 𝑠)})
164fvexi 6857 . . . . . 6 𝐴 ∈ V
1716pwex 5336 . . . . 5 𝒫 𝐴 ∈ V
18 velpw 4566 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 βŠ† 𝐴)
1918anbi1i 625 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠) ↔ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠))
2019abbii 2807 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)} = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)}
21 ssab2 4037 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)} βŠ† 𝒫 𝐴
2220, 21eqsstrri 3980 . . . . 5 {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)} βŠ† 𝒫 𝐴
2317, 22ssexi 5280 . . . 4 {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)} ∈ V
2414, 15, 23fvmpt 6949 . . 3 (𝐾 ∈ V β†’ (PSubClβ€˜πΎ) = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘ )) = 𝑠)})
252, 24eqtrid 2789 . 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 397   = wceq 1542   ∈ wcel 2107  {cab 2714  Vcvv 3446   βŠ† wss 3911  π’« cpw 4561  β€˜cfv 6497  Atomscatm 37728  βŠ₯𝑃cpolN 38368  PSubClcpscN 38400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-psubclN 38401
This theorem is referenced by:  ispsubclN  38403
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