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Theorem lplnset 35316
 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 3350 . 2 (𝐾𝐴𝐾 ∈ V)
2 lplnset.p . . 3 𝑃 = (LPlanes‘𝐾)
3 fveq2 6350 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lplnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2810 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6350 . . . . . . 7 (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7 lplnset.n . . . . . . 7 𝑁 = (LLines‘𝐾)
86, 7syl6eqr 2810 . . . . . 6 (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9 fveq2 6350 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 lplnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10syl6eqr 2810 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 4813 . . . . . 6 (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥𝑦𝐶𝑥))
138, 12rexeqbidv 3290 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑥))
145, 13rabeqbidv 3333 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
15 df-lplanes 35286 . . . 4 LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
16 fvex 6360 . . . . . 6 (Base‘𝐾) ∈ V
174, 16eqeltri 2833 . . . . 5 𝐵 ∈ V
1817rabex 4962 . . . 4 {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ∈ V
1914, 15, 18fvmpt 6442 . . 3 (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
202, 19syl5eq 2804 . 2 (𝐾 ∈ V → 𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
211, 20syl 17 1 (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2137  ∃wrex 3049  {crab 3052  Vcvv 3338   class class class wbr 4802  ‘cfv 6047  Basecbs 16057   ⋖ ccvr 35050  LLinesclln 35278  LPlanesclpl 35279 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-iota 6010  df-fun 6049  df-fv 6055  df-lplanes 35286 This theorem is referenced by:  islpln  35317
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