Theorem psubclsetN 40565

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 3477	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		psubclset.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3		fveq2 6869	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		psubclset.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2817	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	sseq2d 3970	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠 ⊆ 𝐴))
7		fveq2 6869	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
8		psubclset.p	. . . . . . . . 9 ⊢ ⊥ = (⊥𝑃‘𝐾)
9	7, 8	eqtr4di 2817	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = ⊥ )
10	9	fveq1d 6871	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘𝑠) = ( ⊥ ‘𝑠))
11	9, 10	fveq12d 6876	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑠)))
12	11	eqeq1d 2766	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠 ↔ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠))
13	6, 12	anbi12d 641	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠) ↔ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)))
14	13	abbidv 2830	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
15		df-psubclN 40564	. . . 4 ⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)})
16	4	fvexi 6883	. . . . . 6 ⊢ 𝐴 ∈ V
17	16	pwex 5339	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
18		velpw 4562	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 ⊆ 𝐴)
19	18	anbi1i 633	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠) ↔ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠))
20	19	abbii 2831	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)}
21		ssab2 4034	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
22	20, 21	eqsstrri 3985	. . . . 5 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
23	17, 22	ssexi 5280	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ∈ V
24	14, 15, 23	fvmpt 6977	. . 3 ⊢ (𝐾 ∈ V → (PSubCl‘𝐾) = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
25	2, 24	eqtrid 2811	. 2 ⊢ (𝐾 ∈ V → 𝐶 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
26	1, 25	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝐶 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {cab 2742  Vcvv 3456   ⊆ wss 3906  𝒫 cpw 4557  ‘cfv 6523  Atomscatm 39892  ⊥_𝑃cpolN 40531  PSubClcpscN 40563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-psubclN 40564

This theorem is referenced by:  ispsubclN  40566