Theorem psubclsetN 37877

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 3440	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		psubclset.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3		fveq2 6756	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		psubclset.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2797	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	sseq2d 3949	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠 ⊆ 𝐴))
7		fveq2 6756	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = (⊥𝑃‘𝐾))
8		psubclset.p	. . . . . . . . 9 ⊢ ⊥ = (⊥𝑃‘𝐾)
9	7, 8	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (⊥𝑃‘𝑘) = ⊥ )
10	9	fveq1d 6758	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘𝑠) = ( ⊥ ‘𝑠))
11	9, 10	fveq12d 6763	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑠)))
12	11	eqeq1d 2740	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠 ↔ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠))
13	6, 12	anbi12d 630	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠) ↔ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)))
14	13	abbidv 2808	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
15		df-psubclN 37876	. . . 4 ⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)})
16	4	fvexi 6770	. . . . . 6 ⊢ 𝐴 ∈ V
17	16	pwex 5298	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
18		velpw 4535	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 ⊆ 𝐴)
19	18	anbi1i 623	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠) ↔ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠))
20	19	abbii 2809	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)}
21		ssab2 4008	. . . . . 6 ⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
22	20, 21	eqsstrri 3952	. . . . 5 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
23	17, 22	ssexi 5241	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)} ∈ V
24	14, 15, 23	fvmpt 6857	. . 3 ⊢ (𝐾 ∈ V → (PSubCl‘𝐾) = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
25	2, 24	syl5eq 2791	. 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 1539   ∈ wcel 2108  {cab 2715  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ‘cfv 6418  Atomscatm 37204  ⊥_𝑃cpolN 37843  PSubClcpscN 37875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-psubclN 37876

This theorem is referenced by:  ispsubclN  37878