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Theorem islpln 35134
Description: The predicate "is a lattice plane". (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
islpln (𝐾𝐴 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋)))
Distinct variable groups:   𝑦,𝑁   𝑦,𝐾   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝑃(𝑦)

Proof of Theorem islpln
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 lplnset.b . . . 4 𝐵 = (Base‘𝐾)
2 lplnset.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
3 lplnset.n . . . 4 𝑁 = (LLines‘𝐾)
4 lplnset.p . . . 4 𝑃 = (LPlanes‘𝐾)
51, 2, 3, 4lplnset 35133 . . 3 (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
65eleq2d 2716 . 2 (𝐾𝐴 → (𝑋𝑃𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥}))
7 breq2 4689 . . . 4 (𝑥 = 𝑋 → (𝑦𝐶𝑥𝑦𝐶𝑋))
87rexbidv 3081 . . 3 (𝑥 = 𝑋 → (∃𝑦𝑁 𝑦𝐶𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑋))
98elrab 3396 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋))
106, 9syl6bb 276 1 (𝐾𝐴 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  {crab 2945   class class class wbr 4685  cfv 5926  Basecbs 15904  ccvr 34867  LLinesclln 35095  LPlanesclpl 35096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-lplanes 35103
This theorem is referenced by:  islpln4  35135  lplni  35136  lplnbase  35138  lplnnle2at  35145
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