Theorem psubspset 40625

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 3474	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		psubspset.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
3		fveq2 6882	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		psubspset.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2815	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	sseq2d 3966	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠 ⊆ 𝐴))
7		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
8		psubspset.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
9	7, 8	eqtr4di 2815	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
10	9	oveqd 7434	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 ∨ 𝑞))
11	10	breq2d 5119	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 ∨ 𝑞)))
12		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
13		psubspset.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
14	12, 13	eqtr4di 2815	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
15	14	breqd 5118	. . . . . . . . . 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 3336	. . . . . . 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 3228	. . . . . 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 2828	. . . 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 40384	. . . 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 6896	. . . . . 6 ⊢ 𝐴 ∈ V
24	23	pwex 5349	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
25		velpw 4565	. . . . . . . 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 2829	. . . . . 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 4030	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ⊆ 𝒫 𝐴
29	27, 28	eqsstrri 3981	. . . . 5 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ⊆ 𝒫 𝐴
30	24, 29	ssexi 5291	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ∈ V
31	21, 22, 30	fvmpt 6990	. . 3 ⊢ (𝐾 ∈ V → (PSubSp‘𝐾) = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})
32	2, 31	eqtrid 2809	. 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 2145  {cab 2740  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  lecple 17355  joincjn 18405  Atomscatm 40144  PSubSpcpsubsp 40377

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-psubsp 40384

This theorem is used by:  ispsubsp  40626