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Theorem lplnset 37280
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 (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 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.
StepHypRef Expression
1 elex 3426 . 2 (𝐾𝐴𝐾 ∈ V)
2 lplnset.p . . 3 𝑃 = (LPlanes‘𝐾)
3 fveq2 6717 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lplnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2796 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6717 . . . . . . 7 (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7 lplnset.n . . . . . . 7 𝑁 = (LLines‘𝐾)
86, 7eqtr4di 2796 . . . . . 6 (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9 fveq2 6717 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 lplnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10eqtr4di 2796 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 5064 . . . . . 6 (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥𝑦𝐶𝑥))
138, 12rexeqbidv 3314 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑥))
145, 13rabeqbidv 3396 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
15 df-lplanes 37250 . . . 4 LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
164fvexi 6731 . . . . 5 𝐵 ∈ V
1716rabex 5225 . . . 4 {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ∈ V
1814, 15, 17fvmpt 6818 . . 3 (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
192, 18syl5eq 2790 . 2 (𝐾 ∈ V → 𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
201, 19syl 17 1 (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  wrex 3062  {crab 3065  Vcvv 3408   class class class wbr 5053  cfv 6380  Basecbs 16760  ccvr 37013  LLinesclln 37242  LPlanesclpl 37243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-lplanes 37250
This theorem is referenced by:  islpln  37281
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