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Theorem lplnset 35417
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 3365 . 2 (𝐾𝐴𝐾 ∈ V)
2 lplnset.p . . 3 𝑃 = (LPlanes‘𝐾)
3 fveq2 6375 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lplnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2817 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6375 . . . . . . 7 (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7 lplnset.n . . . . . . 7 𝑁 = (LLines‘𝐾)
86, 7syl6eqr 2817 . . . . . 6 (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9 fveq2 6375 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 lplnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10syl6eqr 2817 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 4820 . . . . . 6 (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥𝑦𝐶𝑥))
138, 12rexeqbidv 3301 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑥))
145, 13rabeqbidv 3344 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
15 df-lplanes 35387 . . . 4 LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
164fvexi 6389 . . . . 5 𝐵 ∈ V
1716rabex 4973 . . . 4 {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ∈ V
1814, 15, 17fvmpt 6471 . . 3 (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
192, 18syl5eq 2811 . 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 1652  wcel 2155  wrex 3056  {crab 3059  Vcvv 3350   class class class wbr 4809  cfv 6068  Basecbs 16132  ccvr 35150  LLinesclln 35379  LPlanesclpl 35380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-lplanes 35387
This theorem is referenced by:  islpln  35418
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