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Theorem islpln 33834
 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 33833 . . 3 (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
65eleq2d 2673 . 2 (𝐾𝐴 → (𝑋𝑃𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥}))
7 breq2 4587 . . . 4 (𝑥 = 𝑋 → (𝑦𝐶𝑥𝑦𝐶𝑋))
87rexbidv 3034 . . 3 (𝑥 = 𝑋 → (∃𝑦𝑁 𝑦𝐶𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑋))
98elrab 3331 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋))
106, 9syl6bb 275 1 (𝐾𝐴 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900   class class class wbr 4583  ‘cfv 5804  Basecbs 15695   ⋖ ccvr 33567  LLinesclln 33795  LPlanesclpl 33796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-lplanes 33803 This theorem is referenced by:  islpln4  33835  lplni  33836  lplnbase  33838  lplnnle2at  33845
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