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Theorem psubspset 38257
Description: The set of projective subspaces in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
Hypotheses
Ref Expression
psubspset.l ≀ = (leβ€˜πΎ)
psubspset.j ∨ = (joinβ€˜πΎ)
psubspset.a 𝐴 = (Atomsβ€˜πΎ)
psubspset.s 𝑆 = (PSubSpβ€˜πΎ)
Assertion
Ref Expression
psubspset (𝐾 ∈ 𝐡 β†’ 𝑆 = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))})
Distinct variable groups:   𝑠,π‘Ÿ,𝐴   π‘ž,𝑝,π‘Ÿ,𝑠,𝐾
Allowed substitution hints:   𝐴(π‘ž,𝑝)   𝐡(𝑠,π‘Ÿ,π‘ž,𝑝)   𝑆(𝑠,π‘Ÿ,π‘ž,𝑝)   ∨ (𝑠,π‘Ÿ,π‘ž,𝑝)   ≀ (𝑠,π‘Ÿ,π‘ž,𝑝)

Proof of Theorem psubspset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3465 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 psubspset.s . . 3 𝑆 = (PSubSpβ€˜πΎ)
3 fveq2 6846 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 psubspset.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2791 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
65sseq2d 3980 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑠 βŠ† (Atomsβ€˜π‘˜) ↔ 𝑠 βŠ† 𝐴))
7 fveq2 6846 . . . . . . . . . . . . 13 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = (joinβ€˜πΎ))
8 psubspset.j . . . . . . . . . . . . 13 ∨ = (joinβ€˜πΎ)
97, 8eqtr4di 2791 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = ∨ )
109oveqd 7378 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (𝑝(joinβ€˜π‘˜)π‘ž) = (𝑝 ∨ π‘ž))
1110breq2d 5121 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (π‘Ÿ(leβ€˜π‘˜)(𝑝(joinβ€˜π‘˜)π‘ž) ↔ π‘Ÿ(leβ€˜π‘˜)(𝑝 ∨ π‘ž)))
12 fveq2 6846 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
13 psubspset.l . . . . . . . . . . . 12 ≀ = (leβ€˜πΎ)
1412, 13eqtr4di 2791 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
1514breqd 5120 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (π‘Ÿ(leβ€˜π‘˜)(𝑝 ∨ π‘ž) ↔ π‘Ÿ ≀ (𝑝 ∨ π‘ž)))
1611, 15bitrd 279 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (π‘Ÿ(leβ€˜π‘˜)(𝑝(joinβ€˜π‘˜)π‘ž) ↔ π‘Ÿ ≀ (𝑝 ∨ π‘ž)))
1716imbi1d 342 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((π‘Ÿ(leβ€˜π‘˜)(𝑝(joinβ€˜π‘˜)π‘ž) β†’ π‘Ÿ ∈ 𝑠) ↔ (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠)))
185, 17raleqbidv 3318 . . . . . . 7 (π‘˜ = 𝐾 β†’ (βˆ€π‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘Ÿ(leβ€˜π‘˜)(𝑝(joinβ€˜π‘˜)π‘ž) β†’ π‘Ÿ ∈ 𝑠) ↔ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠)))
19182ralbidv 3209 . . . . . 6 (π‘˜ = 𝐾 β†’ (βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘Ÿ(leβ€˜π‘˜)(𝑝(joinβ€˜π‘˜)π‘ž) β†’ π‘Ÿ ∈ 𝑠) ↔ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠)))
206, 19anbi12d 632 . . . . 5 (π‘˜ = 𝐾 β†’ ((𝑠 βŠ† (Atomsβ€˜π‘˜) ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘Ÿ(leβ€˜π‘˜)(𝑝(joinβ€˜π‘˜)π‘ž) β†’ π‘Ÿ ∈ 𝑠)) ↔ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))))
2120abbidv 2802 . . . 4 (π‘˜ = 𝐾 β†’ {𝑠 ∣ (𝑠 βŠ† (Atomsβ€˜π‘˜) ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘Ÿ(leβ€˜π‘˜)(𝑝(joinβ€˜π‘˜)π‘ž) β†’ π‘Ÿ ∈ 𝑠))} = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))})
22 df-psubsp 38016 . . . 4 PSubSp = (π‘˜ ∈ V ↦ {𝑠 ∣ (𝑠 βŠ† (Atomsβ€˜π‘˜) ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘Ÿ(leβ€˜π‘˜)(𝑝(joinβ€˜π‘˜)π‘ž) β†’ π‘Ÿ ∈ 𝑠))})
234fvexi 6860 . . . . . 6 𝐴 ∈ V
2423pwex 5339 . . . . 5 𝒫 𝐴 ∈ V
25 velpw 4569 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 βŠ† 𝐴)
2625anbi1i 625 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠)) ↔ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠)))
2726abbii 2803 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))} = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))}
28 ssab2 4040 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))} βŠ† 𝒫 𝐴
2927, 28eqsstrri 3983 . . . . 5 {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))} βŠ† 𝒫 𝐴
3024, 29ssexi 5283 . . . 4 {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))} ∈ V
3121, 22, 30fvmpt 6952 . . 3 (𝐾 ∈ V β†’ (PSubSpβ€˜πΎ) = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))})
322, 31eqtrid 2785 . 2 (𝐾 ∈ V β†’ 𝑆 = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))})
331, 32syl 17 1 (𝐾 ∈ 𝐡 β†’ 𝑆 = {𝑠 ∣ (𝑠 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑠 βˆ€π‘ž ∈ 𝑠 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑠))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3447   βŠ† wss 3914  π’« cpw 4564   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  lecple 17148  joincjn 18208  Atomscatm 37775  PSubSpcpsubsp 38009
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 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-psubsp 38016
This theorem is referenced by:  ispsubsp  38258
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