Theorem psubclsetN 40738

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 = (Atoms‘𝐾)
psubclset.p	⊢ ⊥ = (⊥𝑃‘𝐾)
psubclset.c	⊢ 𝐶 = (PSubCl‘𝐾)

Assertion

Ref	Expression
psubclsetN	⊢ (𝐾 ∈ 𝐵 → 𝐶 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})

Distinct variable groups:   𝐴,𝑠   𝐾,𝑠

Allowed substitution hints:   𝐵(𝑠)   𝐶(𝑠)   ⊥ (𝑠)

Proof of Theorem psubclsetN

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3475	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		psubclset.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		psubclset.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2815	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	sseq2d 3968	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠 ⊆ 𝐴))
7		fveq2 6881	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
8		psubclset.p	. . . . . . . . 9 ⊢ ⊥ = (⊥𝑃‘𝐾)
9	7, 8	eqtr4di 2815	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = ⊥ )
10	9	fveq1d 6883	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘𝑠) = ( ⊥ ‘𝑠))
11	9, 10	fveq12d 6888	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑠)))
12	11	eqeq1d 2764	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠 ↔ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠))
13	6, 12	anbi12d 643	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠) ↔ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)))
14	13	abbidv 2828	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
15		df-psubclN 40737	. . . 4 ⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)})
16	4	fvexi 6895	. . . . . 6 ⊢ 𝐴 ∈ V
17	16	pwex 5350	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
18		velpw 4566	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 ⊆ 𝐴)
19	18	anbi1i 635	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠) ↔ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠))
20	19	abbii 2829	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)}
21		ssab2 4032	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
22	20, 21	eqsstrri 3983	. . . . 5 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
23	17, 22	ssexi 5292	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ∈ V
24	14, 15, 23	fvmpt 6989	. . 3 ⊢ (𝐾 ∈ V → (PSubCl‘𝐾) = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
25	2, 24	eqtrid 2809	. 2 ⊢ (𝐾 ∈ V → 𝐶 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
26	1, 25	syl 18	1 ⊢ (𝐾 ∈ 𝐵 → 𝐶 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {cab 2740  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4561  ‘cfv 6536  Atomscatm 40065  ⊥_𝑃cpolN 40704  PSubClcpscN 40736

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-psubclN 40737

This theorem is used by:  ispsubclN  40739