Theorem lplnset 39531

Description: The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)

Hypotheses

Ref	Expression
lplnset.b	⊢ 𝐵 = (Base‘𝐾)
lplnset.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
lplnset.n	⊢ 𝑁 = (LLines‘𝐾)
lplnset.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
lplnset	⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})

Distinct variable groups:   𝑦,𝑁   𝑥,𝐵   𝑥,𝑦,𝐾

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝑁(𝑥)

Proof of Theorem lplnset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3501	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lplnset.p	. . 3 ⊢ 𝑃 = (LPlanes‘𝐾)
3		fveq2 6906	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lplnset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2795	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6906	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7		lplnset.n	. . . . . . 7 ⊢ 𝑁 = (LLines‘𝐾)
8	6, 7	eqtr4di 2795	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9		fveq2 6906	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		lplnset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	eqtr4di 2795	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 5154	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥))
13	8, 12	rexeqbidv 3347	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥))
14	5, 13	rabeqbidv 3455	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
15		df-lplanes 39501	. . . 4 ⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6920	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5339	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 7016	. . 3 ⊢ (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
19	2, 18	eqtrid 2789	. 2 ⊢ (𝐾 ∈ V → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
20	1, 19	syl 17	1 ⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  {crab 3436  Vcvv 3480   class class class wbr 5143  ‘cfv 6561  Basecbs 17247   ⋖ ccvr 39263  LLinesclln 39493  LPlanesclpl 39494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-lplanes 39501

This theorem is referenced by:  islpln  39532