Theorem psubclsetN 39919

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 3499	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		psubclset.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3		fveq2 6907	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		psubclset.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2793	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	sseq2d 4028	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠 ⊆ 𝐴))
7		fveq2 6907	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
8		psubclset.p	. . . . . . . . 9 ⊢ ⊥ = (⊥𝑃‘𝐾)
9	7, 8	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = ⊥ )
10	9	fveq1d 6909	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘𝑠) = ( ⊥ ‘𝑠))
11	9, 10	fveq12d 6914	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑠)))
12	11	eqeq1d 2737	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠 ↔ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠))
13	6, 12	anbi12d 632	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠) ↔ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)))
14	13	abbidv 2806	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
15		df-psubclN 39918	. . . 4 ⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)})
16	4	fvexi 6921	. . . . . 6 ⊢ 𝐴 ∈ V
17	16	pwex 5386	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
18		velpw 4610	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 ⊆ 𝐴)
19	18	anbi1i 624	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠) ↔ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠))
20	19	abbii 2807	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)}
21		ssab2 4089	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
22	20, 21	eqsstrri 4031	. . . . 5 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
23	17, 22	ssexi 5328	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ∈ V
24	14, 15, 23	fvmpt 7016	. . 3 ⊢ (𝐾 ∈ V → (PSubCl‘𝐾) = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
25	2, 24	eqtrid 2787	. 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 395   = wceq 1537   ∈ wcel 2106  {cab 2712  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  ‘cfv 6563  Atomscatm 39245  ⊥_𝑃cpolN 39885  PSubClcpscN 39917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-psubclN 39918

This theorem is referenced by:  ispsubclN  39920