Theorem psubspset 40559

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 3479	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		psubspset.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
3		fveq2 6888	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		psubspset.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2819	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	sseq2d 3972	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠 ⊆ 𝐴))
7		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
8		psubspset.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
9	7, 8	eqtr4di 2819	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
10	9	oveqd 7440	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 ∨ 𝑞))
11	10	breq2d 5126	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 ∨ 𝑞)))
12		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
13		psubspset.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
14	12, 13	eqtr4di 2819	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
15	14	breqd 5125	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝 ∨ 𝑞) ↔ 𝑟 ≤ (𝑝 ∨ 𝑞)))
16	11, 15	bitrd 282	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟 ≤ (𝑝 ∨ 𝑞)))
17	16	imbi1d 344	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠) ↔ (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠)))
18	5, 17	raleqbidv 3341	. . . . . . 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 3232	. . . . . 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 644	. . . . 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 2832	. . . 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 40318	. . . 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	4	fvexi 6902	. . . . . 6 ⊢ 𝐴 ∈ V
24	23	pwex 5356	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
25		velpw 4572	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 ⊆ 𝐴)
26	25	anbi1i 636	. . . . . . 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)))
27	26	abbii 2833	. . . . . 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))}
28		ssab2 4036	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ⊆ 𝒫 𝐴
29	27, 28	eqsstrri 3987	. . . . 5 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ⊆ 𝒫 𝐴
30	24, 29	ssexi 5298	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ∈ V
31	21, 22, 30	fvmpt 6996	. . 3 ⊢ (𝐾 ∈ V → (PSubSp‘𝐾) = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})
32	2, 31	eqtrid 2813	. 2 ⊢ (𝐾 ∈ V → 𝑆 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})
33	1, 32	syl 18	1 ⊢ (𝐾 ∈ 𝐵 → 𝑆 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2744  ∀wral 3082  Vcvv 3458   ⊆ wss 3908  𝒫 cpw 4567   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  lecple 17342  joincjn 18392  Atomscatm 40078  PSubSpcpsubsp 40311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-psubsp 40318

This theorem is used by:  ispsubsp  40560