Theorem lplnset 37470

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 3440	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		lplnset.p	. . 3 ⊢ 𝑃 = (LPlanes‘𝐾)
3		fveq2 6756	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		lplnset.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2797	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7		lplnset.n	. . . . . . 7 ⊢ 𝑁 = (LLines‘𝐾)
8	6, 7	eqtr4di 2797	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9		fveq2 6756	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10		lplnset.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
11	9, 10	eqtr4di 2797	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
12	11	breqd 5081	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥 ↔ 𝑦𝐶𝑥))
13	8, 12	rexeqbidv 3328	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥))
14	5, 13	rabeqbidv 3410	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
15		df-lplanes 37440	. . . 4 ⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
16	4	fvexi 6770	. . . . 5 ⊢ 𝐵 ∈ V
17	16	rabex 5251	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥} ∈ V
18	14, 15, 17	fvmpt 6857	. . 3 ⊢ (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
19	2, 18	syl5eq 2791	. 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 1539   ∈ wcel 2108  ∃wrex 3064  {crab 3067  Vcvv 3422   class class class wbr 5070  ‘cfv 6418  Basecbs 16840   ⋖ ccvr 37203  LLinesclln 37432  LPlanesclpl 37433

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-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-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-lplanes 37440

This theorem is referenced by:  islpln  37471