Theorem psubspset 34048

Description: The set of projective subspaces in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)

Hypotheses

Ref	Expression
psubspset.l	⊢ ≤ = (le‘𝐾)
psubspset.j	⊢ ∨ = (join‘𝐾)
psubspset.a	⊢ 𝐴 = (Atoms‘𝐾)
psubspset.s	⊢ 𝑆 = (PSubSp‘𝐾)

Assertion

Ref	Expression
psubspset	⊢ (𝐾 ∈ 𝐵 → 𝑆 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})

Distinct variable groups:   𝑠,𝑟,𝐴   𝑞,𝑝,𝑟,𝑠,𝐾

Allowed substitution hints:   𝐴(𝑞,𝑝)   𝐵(𝑠,𝑟,𝑞,𝑝)   𝑆(𝑠,𝑟,𝑞,𝑝)   ∨ (𝑠,𝑟,𝑞,𝑝)   ≤ (𝑠,𝑟,𝑞,𝑝)

Proof of Theorem psubspset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3185	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		psubspset.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
3		fveq2 6103	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		psubspset.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	syl6eqr 2662	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	sseq2d 3596	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠 ⊆ 𝐴))
7		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
8		psubspset.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
9	7, 8	syl6eqr 2662	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
10	9	oveqd 6566	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 ∨ 𝑞))
11	10	breq2d 4595	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 ∨ 𝑞)))
12		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
13		psubspset.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
14	12, 13	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
15	14	breqd 4594	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝 ∨ 𝑞) ↔ 𝑟 ≤ (𝑝 ∨ 𝑞)))
16	11, 15	bitrd 267	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟 ≤ (𝑝 ∨ 𝑞)))
17	16	imbi1d 330	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠) ↔ (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠)))
18	5, 17	raleqbidv 3129	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠) ↔ ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠)))
19	18	2ralbidv 2972	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠) ↔ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠)))
20	6, 19	anbi12d 743	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠)) ↔ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))))
21	20	abbidv 2728	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠))} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})
22		df-psubsp 33807	. . . 4 ⊢ PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠))})
23		fvex 6113	. . . . . . 7 ⊢ (Atoms‘𝐾) ∈ V
24	4, 23	eqeltri 2684	. . . . . 6 ⊢ 𝐴 ∈ V
25	24	pwex 4774	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
26		selpw 4115	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 ⊆ 𝐴)
27	26	anbi1i 727	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠)) ↔ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠)))
28	27	abbii 2726	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))}
29		ssab2 3649	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ⊆ 𝒫 𝐴
30	28, 29	eqsstr3i 3599	. . . . 5 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ⊆ 𝒫 𝐴
31	25, 30	ssexi 4731	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ∈ V
32	21, 22, 31	fvmpt 6191	. . 3 ⊢ (𝐾 ∈ V → (PSubSp‘𝐾) = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})
33	2, 32	syl5eq 2656	. 2 ⊢ (𝐾 ∈ V → 𝑆 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})
34	1, 33	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑆 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  Atomscatm 33568  PSubSpcpsubsp 33800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-psubsp 33807

This theorem is referenced by:  ispsubsp  34049