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Theorem lplnset 36841
 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 3459 . 2 (𝐾𝐴𝐾 ∈ V)
2 lplnset.p . . 3 𝑃 = (LPlanes‘𝐾)
3 fveq2 6645 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lplnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2851 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6645 . . . . . . 7 (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7 lplnset.n . . . . . . 7 𝑁 = (LLines‘𝐾)
86, 7eqtr4di 2851 . . . . . 6 (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9 fveq2 6645 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 lplnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10eqtr4di 2851 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 5041 . . . . . 6 (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥𝑦𝐶𝑥))
138, 12rexeqbidv 3355 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑥))
145, 13rabeqbidv 3433 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
15 df-lplanes 36811 . . . 4 LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
164fvexi 6659 . . . . 5 𝐵 ∈ V
1716rabex 5199 . . . 4 {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ∈ V
1814, 15, 17fvmpt 6745 . . 3 (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
192, 18syl5eq 2845 . 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 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110  Vcvv 3441   class class class wbr 5030  ‘cfv 6324  Basecbs 16477   ⋖ ccvr 36574  LLinesclln 36803  LPlanesclpl 36804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-lplanes 36811 This theorem is referenced by:  islpln  36842
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